Statement (i) follows immediately from the \({\beta }\) -triangle inequalities
\(\Vert  h_n\Vert _{F_n}^\beta -\Vert  g_n\Vert _{F_n}^\beta \le \Vert h_n+g_n\Vert _{F_n}^{\beta }\le \Vert  h_n\Vert _{F_n}^\beta +\Vert  g_n\Vert _{F_n}^\beta ,\qquad n\in {\mathbb {N}},\) 
while statements (ii) and (iii) are easy consequences of (i).\(\Box \)
Definition 2.4 
Let \(F_n\)  be equipped with two "norms" \(\Vert \cdot \Vert _{F_n}\)  and
\(\Vert \cdot \Vert _{F_n}^*,\,n\in {\mathbb {N}}\) .
We say that \(\left\lbrace \Vert \cdot \Vert _{F_n}\right\rbrace _{n=1}^{\infty }\)  possesses the GC-property
with respect to \(\left\lbrace \Vert \cdot \Vert _{F_n}^*\right\rbrace _{n=1}^{\infty }\)  if for any
\(h_n\in F_n,\,g_n\in F_n,\,n\in {\mathbb {N}}\) , such that
\(\lim _{n\rightarrow {\infty }}\Vert g_n\Vert _{F_n}^*=0\) ,
relations (REF ) and () hold true.

In case of \(\Vert \cdot \Vert _{F_n} =\Vert \cdot \Vert _{F_n}^*,\,n\in {\mathbb {N}}\) ,
the GC-property is reduced to the C-property. In most of our
applications we use sequences of
"norms" with the C-property. However, in the following
typical example (related to approximation theory)
we present a sequence of "norms" that possesses the GC-property but does
not possess the C-property.
Example 2.5 
Let \(F_n\ne \lbrace 0\rbrace \)  and let \(\Vert \cdot \Vert _{F_n}^*,\,n\in {\mathbb {N}},\)  satisfy property (b)
of Definition REF .
Next, let
\(\left\lbrace \Vert \cdot \Vert _{F_n}^*\right\rbrace _{n=1}^{\infty }\)  possess the C-property. Let us set
\(\Vert f\Vert _{F_n}:=\Vert {\varphi }_n-f\Vert _{F_n}^*,\qquad f\in F_n,\) 
where \({\varphi }_n\)  is a fixed element of \(F_n,\,n\in {\mathbb {N}}\) .
Then \(\left\lbrace \Vert \cdot \Vert _{F_n}\right\rbrace _{n=1}^{\infty }\)  possesses the GC-property
with respect to \(\left\lbrace \Vert \cdot \Vert _{F_n}^*\right\rbrace _{n=1}^{\infty }\) .
Indeed, for any
\(h_n\in F_n,\,g_n\in F_n,\,n\in {\mathbb {N}}\) , such that
\(\lim _{n\rightarrow {\infty }}\Vert g_n\Vert _{F_n}^*=\lim _{n\rightarrow {\infty }}\Vert -g_n\Vert _{F_n}^*=0,\) 
we have
\(\liminf _{n\rightarrow {\infty }}\Vert h_n+g_n\Vert _{F_n}&=&\liminf _{n\rightarrow {\infty }}(\Vert {\varphi }_n-h_n)+(-g_n)\Vert _{F_n}^*\nonumber \\&=&\liminf _{n\rightarrow {\infty }}\Vert {\varphi }_n-h_n\Vert _{F_n}^*\nonumber \\&=&\liminf _{n\rightarrow {\infty }}\Vert h_n\Vert _{F_n}.\) 
Similar relations hold true if we replace \(\liminf \) 
in (REF ) with \(\limsup \) .
Hence (REF ) and () are valid.
On the other hand, since \(F_n\ne \lbrace 0\rbrace ,\,n\in {\mathbb {N}}\) ,
then by property (b) of Definition REF , it is always possible to choose \({\varphi }_n\in F_n,\,n\in {\mathbb {N}}\) ,
such that
\(\lim _{n\rightarrow {\infty }}\Vert {\varphi }_n\Vert _{F_n}^*=a\) , where \(a\in (0,{\infty })\)  is a fixed number.
Then for \(h_n:=(1/3){\varphi }_n\)  and \(g_n:=(1-1/n){\varphi }_n,\,n\in {\mathbb {N}}\) , we have
\(\lim _{n\rightarrow {\infty }}\Vert g_n\Vert _{F_n}=\lim _{n\rightarrow {\infty }}(1/n)\Vert {\varphi }_n\Vert _{F_n}^*=0\) 
and
\(\lim _{n\rightarrow {\infty }}\Vert h_n+g_n\Vert _{F_n}=\lim _{n\rightarrow {\infty }}\Vert {\varphi }_n-h_n-g_n\Vert _{F_n}^*=a/3;\quad \lim _{n\rightarrow {\infty }}\Vert h_n\Vert _{F_n}=\lim _{n\rightarrow {\infty }}\Vert {\varphi }_n-h_n\Vert _{F_n}^*=2a/3.\) 
These relations show that in general \(\left\lbrace \Vert \cdot \Vert _{F_n}\right\rbrace _{n=1}^{\infty }\) 
does not possess the C-property.

Sharp Constants
For vector spaces \(F\)  and \(H\)  equipped with the corresponding "norms" and for
an operator \(L:F\rightarrow H\) , we
define the "norm" of \(L\)  by the formula
\(\Vert L\Vert =\Vert L\Vert _{F\rightarrow H}:=\sup _{h\in F,\,\Vert h\Vert _{F}> 0}\frac{\Vert L(h)\Vert _{H}}{\Vert h\Vert _{F}}.\) 
If \(B\)  is a subset of \(F\) , we
define the "norm" \(\Vert L\Vert \)  of an operator \(L:B\rightarrow H\)  on \(B\)  by the right hand side of
(REF ) with \(h\in F\)  replaced by \(h\in B\) .
Let \(F^{(j)}\ne \lbrace 0\rbrace \)  be a vector space over \({\mathbb {C}}\)  or \({\mathbb {R}}\)  equipped with a "norm"
\(\Vert \cdot \Vert _{F^{(j)}},\,j=1,\,2,\)  and let \(B\)  be a nonempty subset of
\(F^{(2)}\) . Next, let \(D:B\rightarrow F^{(1)}\)  be an operator.
In case of the imbedding operator \(D=I\) , we assume that
\(F^{(1)}\cap F^{(2)}\ne \emptyset \)  and
\(B\subseteq F^{(1)}\cap F^{(2)}\) .
Certainly, if \(F^{(1)}= F^{(2)}\)  and \(\Vert \cdot \Vert _{F^{(1)}}=\Vert \cdot \Vert _{F^{(2)}}\) ,
then \(I\)  is the identity operator.
Let us define the following sharp constant:
\((D,B,F^{(1)},F^{(2)}:=\sup _{f\in B,\,\Vert f\Vert _{F^{(2)}}> 0}\frac{\Vert D(f)\Vert _{F^{(1)}}}{\Vert f\Vert _{F^{(2)}}},\) 
which is the "norm" of \(D\) 
on \(B\) . Clearly, \([0,{\infty }]\) .
In certain cases the following formulae for \( are valid:\begin{eqnarray}&&\sup _{f\in B,\,\Vert f\Vert _{F^{(2)}}=1}\Vert D(f)\Vert _{F^{(1)}}, \\&&\sup _{f\in B,\,\Vert f\Vert _{F^{(2)}}>0,\,\Vert D(f)\Vert _{F^{(1)}}=1}\frac{1}{\Vert f\Vert _{F^{(2)}}}.\end{eqnarray}In particular, it is easy to verify the following proposition.\begin{proposition}(i) Let B be a subspace of F^{(2)},\, D be a 1-homogeneous operator(that is, D({\alpha }f)={\alpha }D(f),\,{\alpha }>0), and "norms" \Vert \cdot \Vert _{F^{(j)}},\,j =1,\,2,satisfy the homogeneity property (b). Then relations (\ref {E2.6}) and(\ref {E2.7}) hold true.\\(ii) If \Vert \cdot \Vert _{F^{(2)}}=\Vert \cdot \Vert _{F^{(2)},\mathrm {triv}},then (\ref {E2.6}) holds true.\\(iii) If \Vert \cdot \Vert _{F^{(1)}}=\Vert \cdot \Vert _{F^{(1)},\mathrm {triv}},then (\ref {E2.7}) holds true.\end{proposition}\subsection {Sharp Constants in Approximation Theory and Analysis}It turns out that the problem of finding \) (D,B,F(1),F(2))\(includes most classical problemsof approximation theory that are related to sharp constants. In addition, it includescertain analysis problems as well. Below we first discuss threemajor approximation problems A, B, and C as special cases of the problemof finding \) .

A. Inequalities for Approximating Elements.
Let \(F^{(j)}\)  be a \(\beta _j\) -normed functional space, \(j=1,2;\, B\)  be
a nontrivial subspace of \(F^{(2)}\) ; and \(D\)  be a linear (often differential)
operator.
Then \( is the sharp constant invarious inequalities for approximating elementsthat we call\emph {Markov-Bernstein typeinequalities of different metrics.} This is one of the most studied topicsof approximation theory. Several special problems are discussed below.\vspace{10.84006pt}\\\emph {A1. Markov-Bernstein Type Inequalities for\) F(1)=F(2)\(.}\\\emph {A1.1. A. A. Markov and V. A. Markov type inequalities} for the set \) B\(of algebraic polynomials of one or several variables. \\\emph {A1.2. Bernstein type inequalities} for the set \) B\(of trigonometric polynomials, splines, or entire functions of exponential typeof one or several variables.\vspace{10.84006pt}\\\emph {A2. Inequalities of Different Metrics for \) D=I\( and\) F(1)F(2)\(.}\\\emph {A2.1. Nikolskii type inequalities} for the set \) B\(of trigonometric or algebraic polynomials, splines, or entire functions ofexponential typeof one or several variables.\\\emph {A2.2. Schur type inequalities} for a nonweighted space \) F(1)\(,a weighted space \) F(2)\(,and the set \) B\(of trigonometric or algebraic polynomials or entire functions of exponential typeof one or several variables.\\\emph {A2.3. Remez type inequalities} for a weighted space \) F(2)\(when a weight vanishes outside of a measurable setand for the set \) B\(of trigonometric, algebraic, or Müntz polynomials.\) There are numerous publications related to all these inequalities, see,
e.g.,
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}
and references therein.

B. Approximation of Individual Elements.
Let us define the operator \(D:B\rightarrow F^{(1)}\)  arbitrarily with the only restriction that
\(D(f)\ne 0\)  for \(f\in B\)  and let us choose
\(\Vert \cdot \Vert _{F^{(1)}}=\Vert \cdot \Vert _{F^{(1)},\mathrm {triv}}.\) 
We denote by
\(F^{(2)}_*\)  the vector space \(F^{(2)}\)  equipped with a certain \({\beta }_2\) -norm
\(\Vert \cdot \Vert ^{*}_{F^{(2)}}\)  on \(F^{(2)},\,{\beta }_2\in (0,1]\) .
For a fixed element \({\varphi }\in F^{(2)}_*\setminus B\) 
we choose \(\Vert f\Vert _{F^{(2)}}=\Vert {\varphi }-f\Vert ^*_{F^{(2)}},\,f\in F^{(2)}_*\) .
Then
\(\frac{1}{\inf _{f\in B,\,f\ne 0}\Vert {\varphi }-f\Vert ^{*}_{F^{(2)}}}=\frac{1}{E\left({\varphi },B,F_{*}^{(2)}\right)},\) 
where \(E\left({\varphi },B,F_{*}^{(2)}\right)\)  is the error of best approximation by elements
from \(B\)  in the space
\(F^{(2)}_{*}\) .
To have \({\infty }\) , we assume that \(E\left({\varphi },B,F_{*}^{(2)}\right)>0\) .
So the problem of finding \( is reduced to finding sharp or asymptoticallysharp value of \linebreak \) E(,B,F*(2))\(.These problems have attracted much attention in classical and contemporaryapproximation theory, having started with classical works by Chebyshev,A. A. Markov, V. A. Markov, Zolotarev, and Bernstein (see, e.g., \cite {T1963, A1965,DL1993, MMR1994, BE1995}).Much contributions to this topic,including asymptotic relations (\ref {E1.1}) through (\ref {E1.4})and their generalizations,have been madein \cite {B1913, B1938, B1946, B1947(lim), T1963, A1965, G1982, G1991, G1992,G2000, S2003, G2010, CV2010, CL2013}and references therein.\vspace{10.84006pt}\\\emph {C. Approximation on Classes of Elements.}Let \) F(1)=F(2), D=I\(, and let us choose\) 
F(2)=F(2),triv.
\(Let \) F(1)\( be a \) 1\(-seminorm of the form\begin{eqnarray*}\Vert f\Vert _{F^{(1)}}:=E\left(f,G,F^{(1)}_{*}\right)=\inf _{g\in G}\Vert f-g\Vert ^{*}_{F^{(1)}},\qquad f\in F^{(1)}_{*} ,\end{eqnarray*}where \) G\( is a subspace of \) F(1)\( and \) F(1)*\( is a vector space\) F(1)\( equipped with a certain\) 1\(-norm \) *F(1)\(on \) F(1), 1(0,1]\(. Then\) 
fBE(f,G,F(1)*)
\(is the sharp constant of approximation by elements from \) G\( in the\) 1\(-normed space \) F*(1)\( on a class \) B\(.Sharp or asymptotically sharp values of this constant of approximationby trigonometric or algebraic polynomials, splines, and entire functionsof exponential type have been found since the 1930s for various classes,see, e.g.,\cite {AK1937, N1946, B1946(Nik), B1947(const), T1963, A1965, K1971,D1975, K1991, G1992,G2000, G2003, G2018a, G2018b} and references therein.\vspace{10.84006pt}\\We also discuss the following important problem in analysis.\vspace{10.84006pt}\\\emph {D. Norms of Operators.}If \) F(j)\( is a \) j\(-normed space, \) j=1, 2,\(and \) B=F(2)\(, then\) D\( is the "norm" of the operator\) D:F(2)F(1)\(.Finding or estimating \)  is one of the major problems in functional analysis,
Fourier analysis, real and complex analysis, and other areas.
Below we discuss just a few examples of such a problem.

D1. Lebesgue constants in Fourier analysis, see, e.g., [19]}
and [20]}.
D2. Summability methods, see, e.g., [1]}
and [22]}.
D3. Hausdorf-Young-F. Riesz inequalities, see, e.g.,
[19]}.
D4. Embedding and extension operators, see, e.g., [24]}, [25]}, [26]}.
D5. Hilbert and maximal Hardy-Littlewood operators,
see, e.g., [27]}.

Problems and Conditions
Here, we discuss major objectives of this paper and introduce conditions that
are needed for our results.
Problems
Let
\(&&=(D,B,F^{(1)},F^{(2)}:=\sup _{f\in B,\,\Vert f\Vert _{F^{(2)}}> 0}\frac{\Vert D(f)\Vert _{F^{(1)}}}{\Vert f\Vert _{F^{(2)}}},\\&&n:=(D_n,B_n,F^{(1)}_n,F^{(2)}_n:=\sup _{f_n\in B_n,\,\Vert f_n\Vert _{F^{(2)}_n}> 0}\frac{\Vert D_n(f_n)\Vert _{F^{(1)}_n}}{\Vert f_n\Vert _{F^{(2)}_n}}\) 
be two sharp constants over two families of components
\(\left\lbrace D,B,F^{(1)},F^{(2)}\right\rbrace \)  and
\(\left\lbrace D_n,B_n,F^{(1)}_n,F^{(2)}_n\right\rbrace \) 
with a running parameter \(n\in {\mathbb {N}}\)  attached to the second family.
Our goal is to obtain asymptotic relations of the form:
\(&&\liminf _{n\rightarrow {\infty }}n, \\&&\limsup _{n\rightarrow {\infty }}n, \\&& \lim _{n\rightarrow {\infty }}n.\) 
In special cases when \( and \) n\( are reduced to approximationof individualelements or approximation on classes of elements, relations like(\ref {E2.8})--(\ref {E2.10})were obtained by Bernstein \cite {B1946(Nik), B1946(trig), B1946,B1947(const), B1947(lim)}for approximation byunivariate trigonometric or algebraic polynomialsand entire functions of exponential type.The author extended these results to multivariate approximation\cite {G1982, G1991} and to a moregeneral setting \cite {G1992, G2000}.Burenkov and Goldman \cite {BG1983} discussed relations like(\ref {E2.8})--(\ref {E2.10}) between norms oflinear operators on periodic and nonperiodic spaces.Relations like (\ref {E2.8})--(\ref {E2.10}) in Markov-Bernsteintype inequalities ofdifferent metricsfor trigonometric and algebraic polynomials and entire functionsof exponential typewere proved by Taikov \cite {T1965, T1993}, Gorbachev \cite {G2005, Gor2018},Levin and Lubinsky \cite {LL2015a,LL2015b},the author and Tikhonov \cite {GT2017},Dai, Gorbachev, and Tikhonov \cite {DGT2018, DGT2019},and the author\cite {G2017,G2018,G2019a,G2019b}.\) Below we introduce some conditions on families
\(\left\lbrace D,B,F^{(1)},F^{(2)}\right\rbrace \)  and
\(\left\lbrace D_n,B_n,F^{(1)}_n,F^{(2)}_n\right\rbrace ,\,n\in {\mathbb {N}}\) ,
that are needed for the proof of relations (REF )–().

Conditions and their Properties
The first group of conditions is needed for the proof of relation (REF ).

Condition C1). A space \(F^{(1)}_n\)  is equipped with another "norm"
\(\Vert \cdot \Vert _{F^{(1)}_n}^*,\,n\in {\mathbb {N}},\)  and
\(\left\lbrace \Vert \cdot \Vert _{F^{(1)}_n}\right\rbrace _{n=1}^{\infty }\)  possesses the GC-property
with respect to \(\left\lbrace \Vert \cdot \Vert _{F^{(1)}_n}^*\right\rbrace _{n=1}^{\infty }\)  (see Definition REF ).

Condition C2). There exists a sequence of operators
\(L_n^{(1)}:F^{(1)}\rightarrow F_n^{(1)},\,n\in {\mathbb {N}}\) , such that the following two conditions hold:

Condition C3). For every \(f\in B\)  with \(\Vert f\Vert _{F^{(2)}}> 0,\) 
there exists a sequence \(f_n\in B_n,\,n\in {\mathbb {N}}\) ,
such that
\(\limsup _{n\rightarrow {\infty }}\Vert f_n\Vert _{F_n^{(2)}}\le \Vert f\Vert _{F^{(2)}}\) 
and
\(\lim _{n\rightarrow {\infty }}\left\Vert L_n^{(1)}(D(f))-D_n(f_n)\right\Vert _{F_n^{(1)}}^*=0.\) 
Condition C4). For every \(h\in F^{(1)}\) ,
\(\liminf _{n\rightarrow {\infty }}\left\Vert L_n^{(1)}(h)\right\Vert _{F_n^{(1)}}\ge \Vert h\Vert _{F^{(1)}}.\) 
In certain cases a slightly different condition is needed.

Condition C4'). Let
\(B\subseteq F^{(1)}\cap F^{(2)}\)  and
\(D(B)\subseteq B\) .
Then for every \(h\in B\) ,
inequality (REF ) holds true.
A group of stronger conditions marked with asterisks
is given below.

Condition C1*). A space \(F^{(j)}_n\)  is equipped with another "norm"
\(\Vert \cdot \Vert _{F^{(j)}_n}^*,\,n\in {\mathbb {N}},\)  and
\(\left\lbrace \Vert \cdot \Vert _{F^{(j)}_n}\right\rbrace _{n=1}^{\infty }\)  possesses the GC-property
with respect to \(\left\lbrace \Vert \cdot \Vert _{F^{(j)}_n}^*\right\rbrace _{n=1}^{\infty }\) , j=1, 2
(see Definition REF ).

Condition C2*). There exist two sequences of operators
\(L_n^{(j)}:F^{(j)}\rightarrow F_n^{(j)},\,n\in {\mathbb {N}},\, j=1,2\) ,
such that the following two conditions hold:

Condition C3*). For every \(f\in B\)  with \(\Vert f\Vert _{F^{(2)}}> 0,\) 
there exists a sequence \(f_n\in B_n,\,n\in {\mathbb {N}}\) ,
such that
\(&&\lim _{n\rightarrow {\infty }}\left\Vert L_n^{(1)}(D(f))-D_n(f_n)\right\Vert _{F_n^{(1)}}^*=0,\\&&\lim _{n\rightarrow {\infty }}\left\Vert f_n-L_n^{(2)}(f)\right\Vert _{F_n^{(2)}}^*=0.\) 
Condition C4*). For every \(h\in F^{(2)}\)  with \(\Vert h\Vert _{F^{(2)}}>0\) ,
\(\limsup _{n\rightarrow {\infty }}\left\Vert L_n^{(2)}(h)\right\Vert _{F_n^{(2)}}\le \Vert h\Vert _{F^{(2)}}.\) 
Proposition 3.1 
Conditions C1*) through C4*) imply Conditions C1) through C3).

It suffices to show that C1*), C2*), () of C3*), and (REF )
of C4*) imply
(REF ). By C2*), C3*), and C4*), for every \(f\in B\)  there exists a sequence
\(\lbrace f_n\rbrace _{n=1}^{\infty }\)  that satisfies (), where \(\left\lbrace L_n^{(2)}\right\rbrace _{n=1}^{\infty }\) 
satisfies (REF ). Then using C1*), (), and (REF ), we obtain
\(\limsup _{n\rightarrow {\infty }}\Vert f_n\Vert _{F_n^{(2)}}=\limsup _{n\rightarrow {\infty }}\left\Vert L_n^{(2)}(f)+\left(f_n-L_n^{(2)}(f)\right)\right\Vert _{F_n^{(2)}}=\limsup _{n\rightarrow {\infty }}\left\Vert L_n^{(2)}(f)\right\Vert _{F_n^{(2)}}\le \Vert f\Vert _{F^{(2)}}.\) 
Hence (REF ) follows.\(\Box \)

Next, we introduce the second group of conditions that is
needed for the proof of relation
().

Condition C5). A space \(F^{(j)}_n\)  is equipped with another "norm"
\(\Vert \cdot \Vert _{F^{(j)}_n}^*,\,n\in {\mathbb {N}},\)  and
\(\left\lbrace \Vert \cdot \Vert _{F^{(j)}_n}\right\rbrace _{n=1}^{\infty }\)  possesses the GC-property
with respect to \(\left\lbrace \Vert \cdot \Vert _{F^{(j)}_n}^*\right\rbrace _{n=1}^{\infty }\) ,  j=1, 2
(see Definition REF ).

Condition C6). There exist two sequences of operators
\(l_n^{(j)}:F^{(j)}\rightarrow F_n^{(j)},\,n\in {\mathbb {N}},\,j=1,2\) , such that the
following two properties hold:

Condition C6a).
For every \(h\in F^{(1)}\)  with \(\Vert h\Vert _{F^{(1)}}>0\) ,
\(\limsup _{n\rightarrow {\infty }}\left\Vert l_n^{(1)}(h)\right\Vert _{F_n^{(1)}}\le \Vert h\Vert _{F^{(1)}}.\) 
Condition C6b). For every \(h\in F^{(2)}\) ,
\(\liminf _{n\rightarrow {\infty }}\left\Vert l_n^{(2)}(h)\right\Vert _{ F^{(2)}_n}\ge \Vert h\Vert _{F^{(2)}}.\) 
Condition C7).
There is a constant \(C_1\in (0,{\infty })\)  such that
for every \(h\in F^{(2)}\)  with \(\Vert h\Vert _{F^{(2)}}>0\) ,
\(\limsup _{n\rightarrow {\infty }}\left\Vert l_n^{(2)}(h)\right\Vert _{F_n^{(2)}}\le C_1\Vert h\Vert _{F^{(2)}}.\) 
Next, we need one of the following two conditions associated with relations
like () and ().

Condition C8). There is a set
\(B_n^\prime \subseteq \left\lbrace f_n\in B_n:\Vert f_n\Vert _{F^{(2)}_n}=1\right\rbrace \) 
such that
the following relations hold:
\(n=\sup _{f_n\in B_n^\prime }\Vert D_n(f_n)\Vert _{F^{(1)}_n}=\sup _{f_n\in B_n^\prime }\frac{\Vert D_n(f_n)\Vert _{F_n^{(1)}}}{\Vert f_n\Vert _{F_n^{(2)}}},\qquad n\in {\mathbb {N}}.\) 
Condition C9).
There is a set
\(B_n^\prime \subseteq \left\lbrace f_n\in B_n:\Vert f_n\Vert _{F^{(2)}_n}>0,\,\Vert D_n(f_n)\Vert _{ F^{(1)}_n}=1\right\rbrace \) 
such that
the following relations hold:
\(n=\sup _{f_n\in B_n^\prime }\frac{1}{\Vert f_n\Vert _{F^{(2)}_n}}=\sup _{f_n\in B_n^\prime }\frac{\Vert D_n(f_n)\Vert _{F_n^{(1)}}}{\Vert f_n\Vert _{F_n^{(2)}}},\qquad n\in {\mathbb {N}}.\) 
Condition C10). (Compactness condition)
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty },\,f_n\in B_n^\prime \setminus \lbrace 0\rbrace ,\,n\in {\mathbb {N}}\) ,
there exist a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural numbers and an element \(f\in B\) 
such that
\(&&\lim _{k\rightarrow {\infty }}\left\Vert D_{n_k}\left(f_{n_k}\right)-l_{n_k}^{(1)}(D(f))\right\Vert _{F_{n_k}^{(1)}}^*=0,\\&&\lim _{k\rightarrow {\infty }}\left\Vert f_{n_k}-l_{n_k}^{(2)}(f)\right\Vert _{F_{n_k}^{(2)}}^*=0.\) 
Further, we discuss a different group of conditions with a weak compactness
condition that replaces C10).

Condition C11). There exist two families of operators
\(l_{n,s}^{(j)}:F_n^{(j)}\rightarrow F_s^{(j)},\,n\in {\mathbb {N}},\,s\in {\mathbb {N}},\,j=1,2\) ,
such that the following property holds:

Condition C11a).
\(\sup _{n\in {\mathbb {N}},\,s\in {\mathbb {N}}}\left\Vert l_{n,s}^{(2)}\right\Vert \le 1.\) 
We recall that the "norm" of an operator is defined by (REF ).

Condition C12). (Weak compactness condition)
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty },\,f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) ,
there exists \(f\in B\)  such that
for any fixed \(s\in {\mathbb {N}}\) ,
there exists a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural numbers
(\(n_k=n_k(s),\,k\in {\mathbb {N}}\) )
such that
\(&&\lim _{k\rightarrow {\infty }}\left\Vert l_{n_k,s}^{(1)}\left(D_{n_k}\left(f_{n_k}\right)\right)-l_{s}^{(1)}(D(f))\right\Vert _{F_{s}^{(1)}}^*=0,\\&&\lim _{k\rightarrow {\infty }}\left\Vert l_{n_k,s}^{(2)}(f_{n_k})-l_{s}^{(2)}(f)\right\Vert _{F_{s}^{(2)}}^*=0,\) 
and, in addition,
\(\left\Vert D_{n_k}\left(f_{n_k}\right)\right\Vert _{F_{n_k}^{(1)}}=\left\Vert l_{n_k,s}^{(1)}\left(D_{n_k}\left(f_{n_k}\right)\right)\right\Vert _{F_{s}^{(1)}},\quad k\in {\mathbb {N}}.\) 
Condition C13).
There is a constant \(C_2\in (0,{\infty })\)  such that
for every \(h_n\in F_n^{(2)}\) ,
\(\limsup _{n\rightarrow {\infty }}\Vert h_n\Vert _{F_n^{(2)}}\le C_2\limsup _{s\rightarrow {\infty }}\limsup _{n\rightarrow {\infty }}\left\Vert l_{n,s}^{(2)}(h_n)\right\Vert _{F_s^{(2)}}.\) 
In Section REF  we will need modified conditions
C6a), C10), and C12) marked with asterisks below.

Condition C6*a).
There exists a sequence of operators
\(l_n^{(1)}:F^{(1)}\rightarrow F_n^{(1)},\,n\in {\mathbb {N}}\) , such that
for every \(h\in F^{(1)}\)  with \(\Vert h\Vert _{F^{(1)}}>0\) ,
\(\limsup _{n\rightarrow {\infty }}\left\Vert l_n^{(1)}(h)\right\Vert _{F_n^{(1)}}\le \Vert h\Vert _{F^{(1)}}.\) 
Condition C10*a).
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty },\,f_n\in B_n^\prime \setminus \lbrace 0\rbrace ,\,n\in {\mathbb {N}}\) ,
there exists a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural numbers and an element \(f\in B\) 
such that
\(\lim _{k\rightarrow {\infty }}\left\Vert D_{n_k}\left(f_{n_k}\right)-l_{n_k}^{(1)}(D(f))\right\Vert _{F_{n_k}^{(1)}}^*=0.\) 
Condition C10*b). For
any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty },\,f_n\in B_n^\prime \setminus \lbrace 0\rbrace ,\,n\in {\mathbb {N}}\) ,
for
the sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural numbers from Condition C10*a),
and for the element \(f\in B\)  from Condition C10*a),
there exists a sequence of operators
\(l_n^{(2)}:F^{(2)}\rightarrow F_n^{(2)},\,n\in {\mathbb {N}}\) , such that
Conditions C6b) and C7) are satisfied. In addition,
the following relation holds:
\(\lim _{k\rightarrow {\infty }}\left\Vert f_{n_k}-l_{n_k}^{(2)}(f)\right\Vert _{F_{n_k}^{(2)}}^*=0.\) 
Condition C12*a).
There exists a family of operators
\(l_{n,s}^{(1)}:F_n^{(1)}\rightarrow F_s^{(1)},\,n\in {\mathbb {N}},\,s\in {\mathbb {N}}\) , such that
for any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty },\,f_n\in B_n^\prime \setminus \lbrace 0\rbrace ,\,n\in {\mathbb {N}}\) ,
there exists \(f\in B\)  such that
for any fixed \(s\in {\mathbb {N}}\) ,
there exists a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural numbers
(\(n_k=n_k(s),\,k\in {\mathbb {N}}\) )
such that
\(&&\lim _{k\rightarrow {\infty }}\left\Vert l_{n_k,s}^{(1)}\left(D_{n_k}\left(f_{n_k}\right)\right)-l_{s}^{(1)}(D(f))\right\Vert _{F_{s}^{(1)}}^*=0,\\&&\left\Vert D_{n_k}\left(f_{n_k}\right)\right\Vert _{F_{n_k}^{(1)}}=\left\Vert l_{n_k,s}^{(1)}\left(D_{n_k}\left(f_{n_k}\right)\right)\right\Vert _{F_{s}^{(1)}},\quad k\in {\mathbb {N}}.\) 
Condition C12*b). For
any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty },\,f_n\in B_n^\prime \setminus \lbrace 0\rbrace ,\,n\in {\mathbb {N}}\) ,
for
the sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural numbers from Condition C12*a)
and the element \(f\in B\) 
from Condition C12*a),
there exists a sequence of operators
\(l_n^{(2)}:F^{(2)}\rightarrow F_n^{(2)},\,n\in {\mathbb {N}}\) , such that
Conditions C6b) and C7) are satisfied and
there exists a sequence of operators
\(l_{n,s}^{(2)}:F_n^{(2)}\rightarrow F_s^{(2)},\,n\in {\mathbb {N}},\,s\in {\mathbb {N}}\) , such that
Conditions C11a) and C13) are satisfied.
In addition, the following relation holds:
\(\lim _{k\rightarrow {\infty }}\left\Vert l_{n_k,s}^{(2)}(f_{n_k})-l_{s}^{(2)}(f)\right\Vert _{F_{s}^{(2)}}^*=0,\qquad s\in {\mathbb {N}}.\) 
Remark 3.2 
Condition C4') includes inequality (REF )
with a weaker requirement \(h\in B\) ,
so if \(B\subseteq F^{(1)}\cap F^{(2)}\)  and
\(D(B)\subseteq B\) 
(for example, if \(D=I\)  is an imbedding operator),
then Condition C4') can efficiently replace Condition C4),
see Theorem REF  and Corollaries REF ,
REF  (i), REF  (i) for details.

Remark 3.3 
Modified compactness conditions C10*b) and C12*b)
are characterized
by the dependence of operators \(\left\lbrace l_n^{(2)}\right\rbrace _{n\in {\mathbb {N}}}\) 
and \(\left\lbrace l_{n,s}^{(2)}\right\rbrace _{n\in {\mathbb {N}},s\in {\mathbb {N}}}\)  on a
sequence \(\lbrace f_n\rbrace _{n=1}^{\infty }\) 
(\(f_n\in B_n^\prime \setminus \lbrace 0\rbrace ,\,n\in {\mathbb {N}}\) ),
the sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\) , and \(f\in B\) ,
while these operators are defined in C6) and C11)
independently of C10) and C12).

Remark 3.4 
The set \(B_n^\prime ,\,n\in {\mathbb {N}}\) , mentioned in the Compactness Conditions
C10), C12), C10*a), C10*b) C12*a), and C12*b)
is defined in C8) or C9). In other words,
if any of these Compactness Conditions is satisfied,
then either C8) or C9)
must be satisfied as well.
Actually, Conditions C8) and C9) state only the existence
of \(B_n^\prime ,\,n\in {\mathbb {N}}\) . So these sets should be defined for each concrete
problem separately (see Chapters  and ).

Remark 3.5 
Conditions C4*), C6a), C6*a), and C7) can be replaced by stronger
conditions for the "norms" of the corresponding operators. For example, if
the condition \(\limsup _{n\rightarrow {\infty }}\left\Vert L_{n}^{(2)}\right\Vert \le 1\) 
is satisfied, then
C4*) is satisfied as well. However in general, C4*) does not imply this condition.

Several conditions defined above are satisfied for certain special "norms."
Proposition 3.6 
Let \(j=1\)  or \(j=2\)  and let \({\beta }_j\in (0,1]\)  be independent of \(n\in {\mathbb {N}}\) .
Next, let \(F_n^{(j)}\)  be equipped with two "norms"
\(\left\Vert \cdot \right\Vert _{F_{n}^{(j)}}\)  and
\(\left\Vert \cdot \right\Vert _{F_{n}^{(j)}}^*,\,n\in {\mathbb {N}}\) 
(see Conditions C1), C1*), and C5)). In addition, let
\(\left\Vert \cdot \right\Vert _{F_{n}^{(j)}},\,n\in {\mathbb {N}}\) ,
be one of the following "norms:"
a)
\(\left\Vert \cdot \right\Vert _{F_{n}^{(j)}}=\left\Vert \cdot \right\Vert _{F_{n}^{(j)}}^*\) 
is a \({\beta }_j\) -norm or a \({\beta }_j\) -seminorm,
\(n\in {\mathbb {N}}\) .
b)
\(\left\Vert \cdot \right\Vert _{F_{n}^{(j)}}=\left\Vert \cdot \right\Vert _{F_{n}^{(j)}}^*:=\Vert \cdot \Vert _{F_n^{(j)},\mathrm {triv}},\,n\in {\mathbb {N}}\) .
c)
\(\left\Vert f_n\right\Vert _{F_{n}^{(j)}}=\Vert {\varphi }_n-f_n\Vert _{F_n^{(j)}}^*,\,f_n\in F_n^{(j)}\) ,
where \({\varphi }_n \)  is a fixed element
of \(F_n^{(j)}\) , and
\(\Vert \cdot \Vert _{F_n^{(j)}}^*\)  is a
\({\beta }_j\) -norm or a \({\beta }_j\) -seminorm,
\(n\in {\mathbb {N}}\) .

Then the following statements are valid:
(i) If \(j=1\) , then Condition C1) is satisfied.
(ii) If \(j=1,\,2\) , then Conditions C1*) and C5) are satisfied.

In case of "norms" a) and b), the corresponding sequence
\(\left\lbrace \Vert \cdot \Vert _{F_n^{(j)}}\right\rbrace _{n=1}^{\infty }=\left\lbrace \Vert \cdot \Vert _{F_n^{(j)}}^*\right\rbrace _{n=1}^{\infty }\) 
possesses
the C-property by statements (ii) and (iii)
of Proposition REF .
In case of "norm" c), the sequence
\(\left\lbrace \Vert \cdot \Vert ^*_{F_n^{(j)}}\right\rbrace _{n=1}^{\infty }\)  possesses
the C-property by statement (ii)
of Proposition REF . Then
\(\left\lbrace \Vert \cdot \Vert _{F_n^{(j)}}\right\rbrace _{n=1}^{\infty }\)  possesses
the GC-property with respect to
\(\left\lbrace \Vert \cdot \Vert ^*_{F_n^{(j)}}\right\rbrace _{n=1}^{\infty }\) .
Indeed, like in Example REF ,
for any
\(h_n\in F_n^{(j)},\,g_n\in F_n^{(j)},\,n\in {\mathbb {N}}\) , such that
\(\lim _{n\rightarrow {\infty }}\Vert g_n\Vert _{F_n^{(j)}}^*=\lim _{n\rightarrow {\infty }}\Vert -g_n\Vert _{F_n^{(j)}}^*=0,\) 
we have
\(\liminf _{n\rightarrow {\infty }}\Vert h_n+g_n\Vert _{F_n^{(j)}}&=&\liminf _{n\rightarrow {\infty }}(\Vert {\varphi }_n-h_n)+(-g_n)\Vert _{F_n^{(j)}}^*\nonumber \\&=&\liminf _{n\rightarrow {\infty }}\Vert {\varphi }_n-h_n\Vert _{F_n^{(j)}}^*\nonumber \\&=&\liminf _{n\rightarrow {\infty }}\Vert h_n\Vert _{F_n^{(j)}}.\) 
Similar relations hold true if we replace \(\liminf \) 
in (REF ) with \(\limsup \) .
Hence (REF ) and ()
are valid for \(F_n\)  replaced with \(F_n^{(j)}\) .
\(\Box \)
Proposition 3.7 
(i) Let \(F_{n}^{(j)},\,n\in {\mathbb {N}}\) , be a \({\beta }_j\) -normed or
\({\beta }_j\) -seminormed space, where
\({\beta }_j\)  is independent of \(n,\,j=1,2.\) 
In addition, let \(B_n\)  be a subspace of \(F_{n}^{(2)}\) 
and let \(D_n\)  be a 1-homogeneous operator
(that is, \(D_n({\alpha }f)={\alpha }D_n(f),\,{\alpha }>0\) ),
\(n\in {\mathbb {N}}\) . Then Conditions C8) and C9)
are satisfied.
(ii) If \(\left\Vert \cdot \right\Vert _{F_{n}^{(2)}}=\Vert \cdot \Vert _{F_n^{(2)},\mathrm {triv}},\,n\in {\mathbb {N}}\) ,
then C8) is satisfied.
(iii) If \(\left\Vert \cdot \right\Vert _{F_{n}^{(1)}}=\Vert \cdot \Vert _{F_n^{(1)},\mathrm {triv}},\,n\in {\mathbb {N}}\) ,
then C9) is satisfied.

Let us define a set \(G_n,\,n\in {\mathbb {N}}\) , by
\(G_n:=\left\lbrace \begin{array}{ll}\left\lbrace f_n\in B_n:\Vert f_n\Vert _{F^{(2)}_n}=1\right\rbrace  & \mbox{for Condition C8)},\\\left\lbrace f_n\in B_n:\Vert f_n\Vert _{F^{(2)}_n}>0,\,\Vert D_n(f_n)\Vert _{ F^{(1)}_n}=1\right\rbrace &\mbox{for Condition C9)}.\end{array}\right.\) 
It follows immediately from Proposition  that relation
(REF ) or relation (REF ) holds true
with \(B_n^\prime \)  replaced by \(G_n,\,n\in {\mathbb {N}}\) .
Therefore, there exists \(B_n^\prime \subseteq G_n,\,n\in {\mathbb {N}}\) , such that
(REF ) or (REF ) is valid
for all premises of statements (i), (ii), and (iii) of Proposition REF 
\(\Box \)
Asymptotic Relations between Sharp Constants in Vector Spaces
Here, we prove
asymptotic relations
(REF ), (), and ().
We first discuss asymptotic relations of the form
\((D,B,F^{(1)},F^{(2)}\le \liminf _{n\rightarrow {\infty }}(D_n,B_n,F^{(1)}_n,F^{(2)}_n=\liminf _{n\rightarrow {\infty }}n.\) 
Theorem 4.1 
If Conditions C1) through C4) are satisfied, then (REF ) holds true.

Let \(f\in B\)  be a fixed element with \(\Vert f\Vert _{F^{(2)}}> 0\) .
Then \(D(f)\in F^{(1)}\)  and using Condition C2) and
inequality (REF ) of Condition C4),
we have
\(\Vert D(f)\Vert _{F^{(1)}}\le \liminf _{n\rightarrow {\infty }}\left\Vert L_n^{(1)}(D(f))\right\Vert _{F_n^{(1)}}.\) 
Next by C2) and C3), there exists a sequence \(f_n\in B_n, n\in {\mathbb {N}},\) 
such that (REF ) and (REF ) hold. Then using C1), we obtain from
(REF ), (REF ), and (REF )
\(&&\Vert D(f)\Vert _{F^{(1)}}\le \liminf _{n\rightarrow {\infty }}\left\Vert D_n(f_n)+\left(L_n^{(1)}(D(f))-D_n(f_n)\right)\right\Vert _{F_n^{(1)}}=\liminf _{n\rightarrow {\infty }}\left\Vert D_n(f_n)\right\Vert _{F_n^{(1)}}\\&&\le \liminf _{n\rightarrow {\infty }}\left(\Vert f_n\Vert _{F^{(2)}_n} n\right)\le \limsup _{n\rightarrow {\infty }}\Vert f_n\Vert _{F^{(2)}_n} \liminf _{n\rightarrow {\infty }}n\le \Vert f\Vert _{F^{(2)}} \liminf _{n\rightarrow {\infty }}n.\) 
Therefore, \(\Vert D(f)\Vert _{F^{(1)}}/\Vert f\Vert _{F^{(2)}}\le \liminf _{n\rightarrow {\infty }}n\) 
and (REF ) is established.\(\Box \)

The following theorem can be proved similarly to Theorem REF .
Theorem 4.2 
If Conditions C1) through C3) and C4') are satisfied,
then (REF ) holds true.

Corollary 4.3 
If Conditions C4) (or C4')) and C1*) through C4*)
are satisfied, then (REF ) holds
true.

The corollary follows immediately from Theorems REF ,
REF ,
and Proposition REF .
Next, we discuss asymptotic relations of the form
\((D,B,F^{(1)},F^{(2)}\ge \limsup _{n\rightarrow {\infty }}(D_n,B_n,F^{(1)}_n,F^{(2)}_n=\limsup _{n\rightarrow {\infty }}n.\) 
In the following Theorems REF , REF , and REF 
we take account of Remark REF .
Theorem 4.4 
Let \(n\)  be a finite number for every \(n\in {\mathbb {N}}\) , and let either
Conditions C7), C8) or Condition C9) be satisfied.
In addition, let C5), C6), and C10) be satisfied.
Then (REF ) holds true.

Assume without loss of generality that \({\infty }\) .
Next, we assume that Conditions C7) and C8) are satisfied.
Since \(n<{\infty }\) , we see from C8) that for any fixed \({\varepsilon }>0\) 
there exists \(f_n=f_{n,{\varepsilon }}\in B_n^\prime \)  with \(\Vert f_n\Vert _{F^{(2)}_n}=1\) 
such that
\(n<\frac{\Vert D_n(f_n)\Vert _{F_n^{(1)}}}{\Vert f_n\Vert _{F_n^{(2)}}}+{\varepsilon },\qquad n\in {\mathbb {N}}.\) 
If Condition C9) is satisfied, then for any fixed \({\varepsilon }>0\) 
there exists \(f_n=f_{n,{\varepsilon }}\in B_n^\prime \)  with
\(\Vert f_n\Vert _{F^{(2)}_n}>0\)  and \(\Vert D_n(f_n)\Vert _{F^{(1)}_n}=1\) 
such that (REF ) holds true as well.
Let \(\lbrace n_p\rbrace _{p=1}^{\infty }\)  be a sequence such that
\(\limsup _{n\rightarrow {\infty }} n=\lim _{p\rightarrow {\infty }} {n_p}.\) 
Next, we apply Conditions C6), C6a), C6b) and C10) to a sequence
\(\lbrace f_{n_p}\rbrace _{p=1}^{\infty }\) . Then there exist
a subsequence \(\lbrace f_{n_{p_k}}\rbrace _{k=1}^{\infty }\)  and an element \(f\in B\) 
such that
\(&&\lim _{k\rightarrow {\infty }}\left\Vert D_{n_{p_k}}\left(f_{n_{p_k}}\right)-l_{n_{p_k}}^{(1)}(D(f))\right\Vert _{F_{n_{p_k}}^{(1)}}^*=0,\\&&\lim _{k\rightarrow {\infty }}\left\Vert f_{n_{p_k}}-l_{n_{p_k}}^{(2)}(f)\right\Vert _{F_{n_{p_k}}^{(2)}}^*=0,\) 
where \(l_{n}^{(j)},\,n\in {\mathbb {N}},\,j=1,\,2\) , satisfy C6a) and C6b).
In particular, \(\Vert f\Vert _{F^{(2)}}<{\infty }\)  since \(B\subseteq F^{(2)}\) .
If C7) and C8) are satisfied, then we have from (), C5), and
(REF ) of C7)
\(1=\lim _{k\rightarrow {\infty }}\left\Vert f_{n_{p_k}}\right\Vert _{F_{n_{p_k}}^{(2)}}&=&\lim _{k\rightarrow {\infty }}\left\Vert l^{(2)}_{n_{p_k}}(f)+\left(f_{n_{p_k}}-l^{(2)}_{n_{p_k}}(f)\right)\right\Vert _{F_{n_{p_k}}^{(2)}}\\&=&\lim _{k\rightarrow {\infty }}\left\Vert l^{(2)}_{n_{p_k}}(f)\right\Vert _{F_{n_{p_k}}^{(2)}}\le C_1\Vert f\Vert _{F^{(2)}}.\) 
Hence there exist \(k_0\in {\mathbb {N}}\)  and constants \(C_3>0\)  and \(C_4>0\)  such that
\(\inf _{k\ge k_0}\left\Vert l^{(2)}_{n_{p_k}}(f)\right\Vert _{F_{n_{p_k}}^{(2)}}\ge C_3,\qquad \Vert f\Vert _{F^{(2)}}\ge C_4.\) 
If C9) is satisfied, then we have from (REF ) and
Conditions C5), C6a), and C6b)
\(1&=&\lim _{k\rightarrow {\infty }}\left\Vert D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right\Vert _{F_{n_{p_k}}^{(1)}}=\lim _{k\rightarrow {\infty }}\left\Vert l^{(1)}_{n_{p_k}}(D(f))+\left(D(f)-l^{(1)}_{n_{p_k}}(D(f))\right)\right\Vert _{F_{n_{p_k}}^{(1)}}\\&=&\lim _{k\rightarrow {\infty }}\left\Vert l^{(1)}_{n_{p_k}}(D(f))\right\Vert _{F_{n_{p_k}}^{(1)}}\le \Vert D(f)\Vert _{F^{(1)}}\le f\Vert _{F^{(2)}}\le _{k\rightarrow {\infty }}\left\Vert l^{(2)}_{n_{p_k}}(f)\right\Vert _{F_{n_{p_k}}^{(2)}}.\) 
Hence (REF ) holds true in this case as well with \(C_4=1/.\) Next using (REF ) and () again, we obtain
by (REF ), (REF ), and C5),
\(\lim _{k\rightarrow {\infty }} {n_{p_k}}\le \frac{\limsup _{k\rightarrow {\infty }}\left\Vert D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right\Vert _{F_{n_{p_k}}^{(1)}}}{\liminf _{k\rightarrow {\infty }}\left\Vert f_{n_{p_k}}\right\Vert _{F_{n_{p_k}}^{(2)}}}+{\varepsilon }= \frac{\limsup _{k\rightarrow {\infty }}\left\Vert l^{(1)}_{n_{p_k}}(D(f))\right\Vert _{F_{n_{p_k}}^{(1)}}}{\liminf _{k\rightarrow {\infty }}\left\Vert l^{(2)}_{n_{p_k}}(f)\right\Vert _{F_{n_{p_k}}^{(2)}}}+{\varepsilon },\) 
where \(\liminf _{k\rightarrow {\infty }}\left\Vert l^{(2)}_{n_{p_k}}(f)\right\Vert _{F_{n_{p_k}}^{(2)}}>0\) 
by (REF ).
Since \(\Vert f\Vert _{F^{(2)}}> 0\)  by (REF ), it follows from (REF ), (REF ),
and Conditions C6a) and C6b) that
\(\limsup _{n\rightarrow {\infty }} n\le \frac{\Vert D(f)\Vert _{F^{(1)}}}{\Vert f\Vert _{F^{(2)}}}+{\varepsilon }\le {\varepsilon }.\) 
Finally letting \({\varepsilon }\rightarrow 0+\)  in (REF ), we arrive at (REF ).
\(\Box \)
Theorem 4.5 
Let \(n\)  be a finite number for every \(n\in {\mathbb {N}}\) ,
and let either Conditions C7), C8), C13) or
Condition C9) be satisfied.
In addition, let C5), C6), C11), and C12)
be satisfied. Then (REF ) holds true.

The proof is similar to that of Theorem REF .
We can assume without loss of generality that \({\infty }\) .
Next, we assume that Conditions C7), C8),
and C13) are satisfied.
Since \(n<{\infty }\) , we see from C8) that for any fixed \({\varepsilon }>0\) 
there exists \(f_n=f_{n,{\varepsilon }}\in B_n^\prime \)  with \(\Vert f_n\Vert _{F^{(2)}_n}=1\) 
such that
\(n<\frac{\Vert D_n(f_n)\Vert _{F_n^{(1)}}}{\Vert f_n\Vert _{F_n^{(2)}}}+{\varepsilon },\qquad n\in {\mathbb {N}}.\) 
If Condition C9) is satisfied, then for any fixed \({\varepsilon }>0\) 
there exists \(f_n=f_{n,{\varepsilon }}\in B_n^\prime \)  with
\(\Vert f_n\Vert _{F^{(2)}_n}>0\)  and \(\Vert D_n(f_n)\Vert _{F^{(1)}_n}=1\) 
such that (REF ) holds true as well.
Let \(\lbrace n_p\rbrace _{p=1}^{\infty }\)  be a sequence such that
\(\limsup _{n\rightarrow {\infty }} n=\lim _{p\rightarrow {\infty }} {n_p}.\) 
Next, we apply Conditions C6), C6a), C6b), C11), and C12) to a sequence
\(\lbrace f_{n_p}\rbrace _{p=1}^{\infty }\) . Then
there exists an element \(f\in B\)  such that
for any fixed \(s\in {\mathbb {N}}\)  there exists
a subsequence \(\lbrace f_{n_{p_k}}\rbrace _{k=1}^{\infty }\) 
(\(n_{p_k}=n_{p_k}(s),\,k\in {\mathbb {N}}\) )
such that
\(&&\lim _{k\rightarrow {\infty }}\left\Vert l_{n_{p_k},s}^{(1)}\left(D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right)-l_{s}^{(1)}(D(f))\right\Vert _{F_{s}^{(1)}}^*=0,\\&&\lim _{k\rightarrow {\infty }}\left\Vert l_{n_{p_k},s}^{(2)}\left(f_{n_{p_k}}\right)-l_{s}^{(2)}(f)\right\Vert _{F_{s}^{(2)}}^*=0,\) 
and in addition,
\(\left\Vert D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right\Vert _{F_{n_{p_k}}^{(1)}}=\left\Vert l_{n_{p_k},s}^{(1)}\left(D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right)\right\Vert _{F_{s}^{(1)}},\quad k\in {\mathbb {N}}.\) 
Here, \(l_{n}^{(j)}\)  and \(l_{n,s}^{(j)},\,n\in {\mathbb {N}},\,s\in {\mathbb {N}},\,j=1,\,2\) ,
satisfy C6a), C6b), and C11a).
In particular, \(\Vert f\Vert _{F^{(2)}}<{\infty }\)  since \(B\subseteq F^{(2)}\) .
If C7), C8), and C13) are satisfied, then
we have from (REF ) of C13), Condition C5), (),
and (REF ) of C7)
\(1&=&\lim _{k\rightarrow {\infty }}\left\Vert f_{n_{p_k}}\right\Vert _{F_{n_{p_k}}^{(2)}}\le C_2\limsup _{s\rightarrow {\infty }}\limsup _{k\rightarrow {\infty }}\left\Vert l_{n_{p_k},s}^{(2)}(f_{n_{p_k}})\right\Vert _{F_s^{(2)}}\\&=&C_2 \limsup _{s\rightarrow {\infty }}\limsup _{k\rightarrow {\infty }}\left\Vert l^{(2)}_{s}(f)+\left(l_{n_{p_k},s}^{(2)}(f_{n_{p_k}})-l^{(2)}_{s}(f)\right)\right\Vert _{F_s^{(2)}}\\&=& C_2\limsup _{s\rightarrow {\infty }}\left\Vert l^{(2)}_{s}(f)\right\Vert _{F_{s}^{(2)}}\le C_1C_2\Vert f\Vert _{F^{(2)}}.\) 
Hence there exist \(s_0\in {\mathbb {N}}\)  and constants \(C_5>0\)  and
\(C_6=1/(C_1C_2)>0\)  such that
\(\inf _{s\ge s_0}\left\Vert l^{(2)}_{s}(f)\right\Vert _{F_{s}^{(2)}}\ge C_5,\qquad \Vert f\Vert _{F^{(2)}}\ge C_6.\) 
If C9) is satisfied, then
by C5) we obtain from (REF ), (REF ), and Conditions
C6a) and C6b)
\(1&=&\lim _{k\rightarrow {\infty }}\left\Vert D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right\Vert _{F_{n_{p_k}}^{(1)}}=\lim _{k\rightarrow {\infty }}\left\Vert l^{(1)}_{n_{p_k},s}\left(D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right)\right\Vert _{F_{s}^{(1)}}\\&=&\left\Vert l^{(1)}_{n_{p_k},s}\left(D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right)+\left(l_s^{(1)}(D(f))-l^{(1)}_{n_{p_k},s}\left(D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right)\right)\right\Vert _{F^{(1)}_s}\\&=&\left\Vert l_s^{(1)}(D(f))\right\Vert _{F^{(1)}_s}\le \Vert D(f)\Vert _{F^{(1)}}\le f\Vert _{F^{(2)}}.\) 
Hence (REF ) holds true in this case as well
with \(C_6=1/.\) Further, we use (REF ), (REF ), (REF ), (),
and Conditions C5) and C11a) to obtain
the following relations for \(s\ge s_0\) :
\(\lim _{k\rightarrow {\infty }} {n_{p_k}}&\le &\frac{\limsup _{k\rightarrow {\infty }}\left\Vert D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right\Vert _{F_{n_{p_k}}^{(1)}}}{\liminf _{k\rightarrow {\infty }}\left\Vert f_{n_{p_k}}\right\Vert _{F_{n_{p_k}}^{(2)}}}+{\varepsilon }\\&\le & \frac{\limsup _{k\rightarrow {\infty }}\left\Vert l_{n_{p_k},s}^{(1)}\left(D_{n_{p_k}}\left(f_{n_{p_k}}\right)\right)\right\Vert _{F_{s}^{(1)}}}{\liminf _{k\rightarrow {\infty }}\left\Vert l_{n_{p_k},s}^{(2)}\left(f_{n_{p_k}}\right)\right\Vert _{F_{s}^{(2)}}}+{\varepsilon }\\&=&\frac{\left\Vert l^{(1)}_{s}(D(f))\right\Vert _{F_{s}^{(1)}}}{\left\Vert l^{(2)}_{s}(f)\right\Vert _{F_{s}^{(2)}}}+{\varepsilon }.\) 
Therefore for some \(f\in B\) , satisfying (REF ), we have
\(\lim _{k\rightarrow {\infty }} {n_{p_k}}\le \frac{\limsup _{s\rightarrow {\infty }}\left\Vert l^{(1)}_{s}(D(f))\right\Vert _{F_{s}^{(1)}}}{\liminf _{s\rightarrow {\infty }}\left\Vert l^{(2)}_{s}(f)\right\Vert _{F_{s}^{(2)}}}+{\varepsilon },\) 
where \(\liminf _{s\rightarrow {\infty }}\left\Vert l^{(2)}_{s}(f)\right\Vert _{F_{s}^{(2)}}>0\) 
by (REF ).
Since \(\left\Vert f\right\Vert _{F^{(2)}}>0\)  by (REF ),
it follows from
(REF ),
(REF ) and Conditions C6a) and C6b) that
\(\limsup _{n\rightarrow {\infty }} {n}\le \frac{\Vert D(f)\Vert _{F^{(1)}}}{\Vert f\Vert _{F^{(2)}}}+{\varepsilon }\le {\varepsilon }.\) 
Finally letting \({\varepsilon }\rightarrow 0+\)  in (REF ), we arrive at (REF ).
\(\Box \)

Combining Theorems REF , REF , REF 
and Corollary REF , we can obtain the relation
\((D,B,F^{(1)},F^{(2)}= \lim _{n\rightarrow {\infty }}(D_n,B_n,F^{(1)}_n,F^{(2)}_n=\limsup _{n\rightarrow {\infty }}n.\) 
Corollary 4.6 
If all the conditions of Theorems REF 
or REF  or Corollary REF 
are satisfied and, in addition, all the conditions of Theorem REF 
or Theorem REF  are satisfied, then relation (REF ) holds true.

Special versions of Theorems REF 
and REF  are presented below.
Theorem 4.7 
Let \(n\)  be a finite number for every \(n\in {\mathbb {N}}\) .
(a) Let
Conditions C5), C6*a), C8), C10*a), and C10*b) be satisfied.
Then (REF ) holds true.
(b) Let
Conditions C5), C6*a), C8), C12*a), and C12*b) be satisfied.
Then (REF ) holds true.

The proof of Theorem REF  follows that of Theorem
REF  (or Theorem REF ) if Condition C10)
(or Condition C12)) is replaced with Conditions
C10*a) and C10*b) (or Conditions C12*a)
and C12*b)).
The outline of the proof with a new component of the proof,
compared with that of Theorem REF 
(or Theorem REF ), is presented below.
Given \({\varepsilon }>0\)  let \(\lbrace f_n\rbrace _{n=1}^{\infty },\,f_n=f_{n,{\varepsilon }}\in B_n^\prime \)  with \(\Vert f_n\Vert _{F^{(2)}_n}=1\) 
be a sequence such that (REF ) (or (REF ))
holds true.
In addition, let \(\lbrace n_p\rbrace _{p=1}^{\infty }\)  be a sequence such that
(REF ) (or (REF )) is valid.
Next, by Condition C10*a) (or C12*a), there exist a subsequence
\(\lbrace n_{p_k}\rbrace _{k=1}^{\infty }\)  and \(f\in B\)  such that
(REF ) (or (REF ) and (REF )) hold(s) true.
Further, by Condition C10*b) (or C12*b), there exists a sequence
of operators \(\left\lbrace l_n^{(2)}\right\rbrace _{n=1}^{\infty }\) 
(or, in addition, a sequence of operators
\(\left\lbrace l_{n,s}^{(2)}\right\rbrace _{n,s=1}^{\infty }\) )
such that () (or ()) is valid.
The rest of the proof can be copied from the proof
of Theorem REF  (or Theorem REF ).\(\Box \)
Remark 4.8 
It is not difficult to extend asymptotic relations of this section to
families \(D_{\alpha },\,B_{\alpha },\,F_{\alpha }^{(1)},\)  and \(F_{\alpha }^{(2)}\)  defined on a set \(E\) 
filtering with respect to a filter \(\Phi \) . The corresponding upper
and lower limits are considered with respect to the filtering set \(E\) 
(cf. [28]}).

Remark 4.9 
We apply the results of this section to Problems A, B, and C
of approximation theory (see Sections 2.3, 5, and 7).
However, it is possible to apply them to Problem D of
analysis as well (see Section 2.3). The similar condition-based approach
to norms of linear operators in periodic and nonperiodic spaces
was discussed in [25]}.

Asymptotic Relations in Approximation Theory
We recall that three major Problems A, B, and C of
approximation theory related
to sharp constants are described in Section .
Here, we apply Theorems REF , REF 
REF , REF , and REF 
and Corollary REF  to these problems of approximation theory.
In Problem A, in general, we cannot essentially reduce
the number of conditions
in asymptotic results of Section .
However, we can do that in Problems B and C.
Inequalities for Approximating Elements (Problem A)
In this case we assume that
\(F^{(j)}\)  and \(F^{(j)}_n\)  are \(\beta _j\) -normed spaces,
where \({\beta }_j\)  is independent of \(n,\,j=1,2,\,n\in {\mathbb {N}}\) .
In addition, we set
\(\Vert \cdot \Vert _{F^{(j)}_n}^*=\Vert \cdot \Vert _{F^{(j)}_n},\,j=1,2,\,n\in {\mathbb {N}}\) .
Next, let \(B\) 
and \(B_n\)  be
nontrivial subspaces of \(F^{(2)}\)  and \(F^{(2)}_n\) , respectively,
and let \(D_n\)  be a 1-homogeneous operator (that is,
\(D_n({\alpha }f)={\alpha }D_n(f),\,{\alpha }>0\) ), \(n\in {\mathbb {N}}\) .
Then Conditions C1), C1*), and C5) are satisfied by
Proposition REF .
In addition, Conditions C8) and C9) are satisfied
by
Proposition REF  (i).
However, it appears that in Problem A Conditions C1) through C3)
are more efficient than Conditions C1*) through C4*)
and Condition C9)
is more efficient than Condition C8)
(see Corollary REF  and Examples REF ,
REF , and REF ).
The following corollary follows from Theorems
REF , REF ,
REF , and REF .
Corollary 5.1 
(i) If Conditions C2), C3), C4) (or C4'))
are satisfied, then
\(\sup _{f\in B,\,\Vert f\Vert _{F^{(2)}}> 0}\frac{\Vert D(f)\Vert _{F^{(1)}}}{\Vert f\Vert _{F^{(2)}}}\le \liminf _{n\rightarrow {\infty }}\sup _{f\in B_n,\,\Vert f\Vert _{F^{(2)}_n}> 0}\frac{\Vert D_n(f)\Vert _{F^{(1)}_n}}{\Vert f\Vert _{F^{(2)}_n}}.\) 
(ii) Let \(n\) 
be a finite number for every \(n\in {\mathbb {N}}\) , and let
Conditions C6) and C10) be satisfied. Then
\(\sup _{f\in B,\,\Vert f\Vert _{F^{(2)}}> 0}\frac{\Vert D(f)\Vert _{F^{(1)}}}{\Vert f\Vert _{F^{(2)}}}\ge \limsup _{n\rightarrow {\infty }}\sup _{f\in B_n,\,\Vert f\Vert _{F^{(2)}_n}> 0}\frac{\Vert D_n(f)\Vert _{F^{(1)}_n}}{\Vert f\Vert _{F^{(2)}_n}}.\) 
(iii) Let \(n\) 
be a finite number for every \(n\in {\mathbb {N}}\) , and let
Conditions C6), C11), and C12) be satisfied.
Then (REF ) holds
true.

Remark 5.2 
Unlike Problems B and C, inequalities for
approximating elements were not included into
the general approach to limit relations between sharp constants
developed in [30]}, [31]}.


Approximation of Individual Elements
(Problem B)
In this case we set
\(F^{(1)}_n=F^{(1)};\,D(f):=f_0,\,f\in B;\,D_n(f_n):=f_0,\,f_n\in B_n;\,n\in {\mathbb {N}},\) 
where \(f_0\ne 0\)  is
a fixed element of \(F^{(1)}\) . In addition, we set
\(\Vert \cdot \Vert _{F^{(1)}_n}^*=\Vert \cdot \Vert _{F^{(1)}_n}=\Vert \cdot \Vert _{F^{(1)}}:=\Vert \cdot \Vert _{F^{(1)},\mathrm {triv}},\qquad n\in {\mathbb {N}},\) 
where \(\Vert \cdot \Vert _{F^{(1)},\mathrm {triv}}\)  is defined by (REF ).
We also assume that \(B\)  and \(B_n\)  are proper nonempty subsets of
\(F^{(2)}\)  and \(F^{(2)}_n\) , respectively, \(n\in {\mathbb {N}}\) .
It follows from (REF ) and (REF ) that
\(\Vert D(f)\Vert _{F^{(1)}}=\left\Vert D_n(f_n)\right\Vert _{F^{(1)}_n}=1,\qquad f\in B,\quad f_n\in B_n,\quad n\in {\mathbb {N}}.\) 
Next, for certain \({\beta }_2\) -norms \(\Vert \cdot \Vert _{F^{(2)}}^*\)  and
\(\Vert \cdot \Vert _{F^{(2)}_n}^*,\,n\in {\mathbb {N}}\) , where \({\beta }_2\in (0,1]\)  is independent of \(n\) ,
we define the following "norms"
\(\Vert h\Vert _{F^{(2)}}:=\Vert {\varphi }-h\Vert _{F^{(2)}}^*,\quad h\in F^{(2)};\qquad \Vert h_n\Vert _{F^{(2)}_n}:=\Vert {\varphi }_n-h_n\Vert _{F^{(2)}_n}^*,\quad h_n\in F^{(2)}_n,\) 
where \({\varphi }\in F^{(2)}\setminus B\)  and \({\varphi }_n\in F^{(2)}_n\setminus B_n,\,n\in {\mathbb {N}}\) ,
are fixed elements. Then by (REF ), (REF ), and (REF ),
\(1/E\left({\varphi },B,F^{(2)}_*\right),\qquad n=1/E\left({\varphi }_n,B_n,F^{(2)}_{*n}\right),\) 
where \(F^{(2)}_{*}\)  and \(F^{(2)}_{*n}\)  are vector spaces
\(F^{(2)}\)  and \(F^{(2)}_{n}\)  equipped with the "norms" \(\Vert \cdot \Vert _{F^{(2)}}^*\)  and
\(\Vert \cdot \Vert _{F^{(2)}_n}^*,\,n\in {\mathbb {N}}\) , respectively.
In addition, we recall that
\(E\left({\varphi },B,F^{(2)}_*\right)=\inf _{f \in B}\Vert {\varphi }-f\Vert ^*_{F^{(2)}},\quad E\left({\varphi }_n,B_n,F^{(2)}_{*n}\right)=\inf _{f_n \in B_n}\Vert {\varphi }_n-f_n\Vert ^*_{F^{(2)}_n},\quad n\in {\mathbb {N}},\) 
are the errors of best approximation (see (REF )).
We also assume that
\(E\left({\varphi },B,F^{(2)}_*\right)>0\)  and
\(E\left({\varphi }_n,B_n,F^{(2)}_{*n}\right)>0\) , so \({\infty }\) 
and \(n<{\infty },\,n\in {\mathbb {N}}\) .
Further, let
\(L_n^{(1)}=l_n^{(1)}=l_{n,s}^{(1)}=I,\,n\in {\mathbb {N}},\,s\in {\mathbb {N}},\) 
be the corresponding identity
operators (because of (REF )
and (REF )).
Since by (REF ),
\(B_n\setminus \lbrace 0\rbrace =\left\lbrace f_n\in B_n:\Vert f_n\Vert _{F^{(2)}_n}>0,\,\Vert D_n(f_n)\Vert _{ F^{(1)}_n}=1\right\rbrace \) ,
we choose \(B_n^\prime \) 
from Condition C9) as a subset of \(B_n\setminus \lbrace 0\rbrace \) ,
satisfying the equality
\(E\left({\varphi }_n,B_n,F^{(2)}_{*n}\right)=E\left({\varphi }_n,B_n^\prime ,F^{(2)}_{*n}\right),\qquad n\in {\mathbb {N}}.\) 
Then Conditions C1), C1*), and C5) are satisfied by
Proposition REF . In addition, Condition C9) is satisfied
by the choice of \(B_n^\prime ,\,n\in {\mathbb {N}}\) , and by Proposition REF  (iii)
if we take into account (REF ), (REF ), and (REF ).
Next, relations
(REF ), (REF ), (REF ), (REF ), (REF ),
(REF ), and (REF )
of Conditions C3), C3*), C4), C4'), C6a), C10),
and C12) are trivially satisfied by (REF ), (REF ),
(REF ), and (REF ).
In particular, Conditions C1) through C3) can be reduced to
inequality (REF ) from C3),
which coincides with (REF ) below.
Since (REF ) is one of the inequalities we wish to prove,
Conditions C1) through C3) are obviously inefficient in
Problem B.
However, Conditions C1*) through C4*)
can be reduced to more efficient relations () and (REF )
from C3*) and C4*), respectively
(see Corollary REF  (i) and Examples REF ,
REF , and REF ).
The following corollary follows from relations
(REF ) through (REF ), Corollary REF , and Theorems
REF  and REF .
Corollary 5.3 
(i) If there exists a sequence of operators
\(L_n^{(2)}:F^{(2)}\rightarrow F_n^{(2)},\,n\in {\mathbb {N}}\) , such that
relations () of C3*) and (REF ) of C4*)
are satisfied, then
\(\limsup _{n\rightarrow {\infty }} E\left({\varphi }_n,B_n,F^{(2)}_{*n}\right)\le E\left({\varphi },B,F^{(2)}_*\right).\) 
(ii) If there exists a sequence of operators
\(l_n^{(2)}:F^{(2)}\rightarrow F_n^{(2)},\,n\in {\mathbb {N}}\) , such that
C6b) and relation () of C10) are satisfied, then
\(\liminf _{n\rightarrow {\infty }} E\left({\varphi }_n,B_n,F^{(2)}_{*n}\right)\ge E\left({\varphi },B,F^{(2)}_*\right).\) 
(iii) If there exist families of operators
\(l_n^{(2)}:F^{(2)}\rightarrow F_n^{(2)},\,n\in {\mathbb {N}}\) ,
and \(l_{n,s}^{(2)}:F^{(2)}_n\rightarrow F_s^{(2)},\,n\in {\mathbb {N}},\,s\in {\mathbb {N}}\) ,
such that
C6b), C11a), and relation () of C12) are satisfied, then
(REF ) holds true.

Remark 5.4 
Special cases of Corollary REF  were discussed in [30]}, [31]}.
In case of linear operators \(L_n^{(2)}:F_{*}^{(2)}\rightarrow F_{*n}^{(2)}\) 
with
\(\limsup _{n\rightarrow {\infty }}\left\Vert L_{n}^{(2)}\right\Vert _{F_{*}^{(2)}\rightarrow F_{*n}^{(2)}}\le 1\)  and \({\varphi }_n=L_n^{(2)}({\varphi }),\,n\in {\mathbb {N}}\) , relation (REF )
of C4*) is satisfied. Indeed, for every \(h\in F^{(2)}\) ,
\(\limsup _{n\rightarrow {\infty }}\left\Vert L_{n}^{(2)}(h)\right\Vert _{F_{n}^{(2)}}=\limsup _{n\rightarrow {\infty }}\left\Vert L_{n}^{(2)}({\varphi })-L_{n}^{(2)}(h)\right\Vert _{F_{n}^{(2)}}^*\le \Vert {\varphi }-h\Vert ^*_{F^{(2)}}=\Vert h\Vert _{F^{(2)}}.\) 
Next, in case of linear operators \(l_n^{(2)}:F_{*}^{(2)}\rightarrow F_{*n}^{(2)}\) 
with
\(\liminf _{n\rightarrow {\infty }}\left\Vert l_{n}^{(2)}(h)\right\Vert ^*_{F_{n}^{(2)}}\ge \Vert h\Vert ^*_{F^{(2)}}\)  and \({\varphi }_n=l_n^{(2)}({\varphi }),\,n\in {\mathbb {N}}\) , relation (REF )
of C6b) is satisfied. The proof of this fact is similar to (REF ).
Further, in case of linear operators \(l_{n,s}^{(2)}:F_{*n}^{(2)}\rightarrow F_{*s}^{(2)}\) 
with
\(l_s^{(2)}=l_{n,s}^{(2)}\,l_n^{(2)},\,n\in {\mathbb {N}},\,s\in {\mathbb {N}};\linebreak \sup _{n\in {\mathbb {N}},s\in {\mathbb {N}}}\left\Vert l_{n,s}^{(2)}\right\Vert _{F_{*n}^{(2)}\rightarrow F_{*s}^{(2)}}\le 1\) ,
and \({\varphi }_n=l_n^{(2)}({\varphi }),\,n\in {\mathbb {N}}\) , relation (REF )
of C11a) is satisfied.
Indeed, for every \(h\in F^{(2)}\) ,
\(\left\Vert l_{n,s}^{(2)}(h)\right\Vert _{F_{s}^{(2)}}=\left\Vert l_{s}^{(2)}({\varphi })-l_{n,s}^{(2)}(h)\right\Vert _{F_{s}^{(2)}}^*=\left\Vert l_{n,s}^{(2)}\left(l_{n}^{(2)}({\varphi })-h\right)\right\Vert _{F_{s}^{(2)}}^*\le \Vert l_{n}^{(2)}({\varphi })-h\Vert ^*_{F^{(2)}_n}=\Vert h\Vert _{F^{(2)}_n}.\) 
In all these special cases,
Corollary REF  was established in [30]} for
normed spaces \(F_{*}^{(2)}\)  and \(F_{*n}^{(2)},\,n\in {\mathbb {N}}\) . For
\({\beta }_2\) -normed spaces
\(F_{*}^{(2)}\)  and \(F_{*n}^{(2)},\,n\in {\mathbb {N}}\) , the corresponding results were proved in
[31]}.


Approximation on Classes of Elements (Problem C)
We set
\(F^{(2)}=F^{(1)},\quad F^{(2)}_n=F^{(1)}_n,\quad D=D_n=I,\quad \,n\in {\mathbb {N}},\) 
where \(I\)  is the corresponding imbedding operator,
and
\(\Vert \cdot \Vert _{F^{(2)}}:=\Vert \cdot \Vert _{F^{(2)},\mathrm {triv}},\qquad \Vert \cdot \Vert _{F^{(2)}_n}^*=\Vert \cdot \Vert _{F^{(2)}_n}:=\Vert \cdot \Vert _{F^{(2)}_n,\mathrm {triv}},\quad n\in {\mathbb {N}}.\) 
It follows from (REF ) that
\(\Vert f\Vert _{F^{(2)}}=\left\Vert f_n\right\Vert _{F^{(2)}_n}=1,\qquad f\in B\setminus \lbrace 0\rbrace ,\quad f_n\in B_n\setminus \lbrace 0\rbrace ,\quad n\in {\mathbb {N}}.\) 
Next, for \({\beta }_1\) -norms \(\Vert \cdot \Vert _{F^{(1)}}^*\)  and
\(\Vert \cdot \Vert _{F^{(1)}_n}^*,\,n\in {\mathbb {N}}\) , where \({\beta }_1\in (0,1]\)  is independent of \(n\) ,
we define the following "norms"
\(&&\Vert h\Vert _{F^{(1)}}:=E\left(h,G,F^{(1)}_*\right)=\inf _{g\in G}\Vert h-g\Vert ^*_{F^{(1)}},\qquad h\in F^{(1)},\\&&\Vert h_n\Vert _{F^{(1)}_n}:=E\left(h_n,G_n,F^{(1)}_{*n}\right)=\inf _{g_n\in G_n}\Vert h_n-g_n\Vert ^*_{F^{(1)}_n},\qquad h_n\in F^{(1)}_n,\quad n\in {\mathbb {N}}.\) 
Here, \(F^{(1)}_{*}\)  and \(F^{(1)}_{*n}\)  are vector spaces
\(F^{(1)}\)  and \(F^{(1)}_{n}\)  equipped with the "norms" \(\Vert \cdot \Vert _{F^{(1)}}^*\)  and
\(\Vert \cdot \Vert _{F^{(1)}_n}^*\) , respectively, and, in addition,
\(G\)  and \(G_n\)  are subspaces of \(F^{(1)}_{*}\)  and \(F^{(1)}_{*n}\) ,
respectively,  \(n\in {\mathbb {N}}\) .
Note that
\(\Vert \cdot \Vert _{F^{(1)}}\)  and \(\Vert \cdot \Vert _{F^{(1)}_n},\,n\in {\mathbb {N}}\) ,
defined by (REF ) and ()
are \({\beta }_1\) -seminorms.
Then by (REF ), (REF ), (REF ), and (),
\(\sup _{f\in B\setminus \lbrace 0\rbrace }E\left(f,G,F^{(1)}_*\right),\quad n=\sup _{f_n\in B_n\setminus \lbrace 0\rbrace }E\left(f_n,G_n,F^{(1)}_{*n}\right)=\sup _{f_n\in B_n}E\left(f_n,G_n,F^{(1)}_{*n}\right)\) 
are sharp constants of best approximation on classes of elements
\(B\setminus \lbrace 0\rbrace \)  and \(B_n,\,n\in {\mathbb {N}}\) .
In addition,
since by (REF ),
\(B_n\setminus \lbrace 0\rbrace =\left\lbrace f_n\in B_n:\Vert f_n\Vert _{ F^{(2)}_n}=1\right\rbrace \) , we
choose \(B_n^\prime \)  from Condition C8) as a subset of \(B_n\setminus \lbrace 0\rbrace \) ,
satisfying the equality
\(\sup _{f_n\in B_n}E\left(f_n,G_n,F^{(1)}_{*n}\right)=\sup _{f_n\in B_n^\prime }E\left(f_n,G_n,F^{(1)}_{*n}\right), n\in {\mathbb {N}}.\) 
Let us also define \(l_n^{(2)}:F^{(2)}\rightarrow F_n^{(2)}\) 
and \(l_{n,s}^{(2)}:F^{(2)}_n\rightarrow F_s^{(2)}\) 
for each sequence \(\lbrace f_n\rbrace _{n=1}^{\infty },\,f_n\in B_n\setminus \lbrace 0\rbrace ,\,n\in {\mathbb {N}},\,s\in {\mathbb {N}}\) , and \(f\in B\) 
(mentioned in C10*a)
and C12*a)), by the formulae
\(l_n^{(2)}(h):=\left\lbrace \begin{array}{ll}f_n,&h=f,\\f_{n,0},&h\in F^{(2)}\setminus \lbrace f\rbrace , \end{array}\right.\quad l_{n,s}^{(2)}(h_n):=\left\lbrace \begin{array}{ll}f_s,&h_n=f_n,\\f_{s,0},&h_n\in F^{(2)}_n\setminus \lbrace f_n\rbrace , \end{array}\right.\) 
where \(f_{n,0}\ne 0\)  is
a fixed element from \(F_n^{(2)},\,n\in {\mathbb {N}}\) .
It follows from (REF ) and (REF ) that
for every \(h\in F^{(2)}\setminus \lbrace f\rbrace \)  and every
\(h_n\in F^{(2)}_n\setminus \lbrace f_n\rbrace \) ,
\(\left\Vert l_n^{(2)}(h)\right\Vert _{F_n^{(2)}}=\left\Vert l_{n,s}^{(2)}(h_n)\right\Vert _{F_s^{(2)}}=1;\qquad l_n^{(2)}(f)=f_n,\quad l_{n,s}^{(2)}(f_n)=f_s,\quad n\in {\mathbb {N}},\quad s\in {\mathbb {N}}.\) 
Then Conditions C1), C1*), C5) are satisfied by
Proposition REF . In addition, Condition C8)
is satisfied by the choice of \(B_n^\prime ,\,n\in {\mathbb {N}}\) , and
by Proposition REF  (ii) if we take into account (REF ),
(REF ), and (REF ).
Next, relations
(REF ), (),
(REF ), (REF ), (REF ) (with \(C_1=1\) ),
(),
(REF ), (), (REF ) (with \(C_2=1\) ),
(REF ), and (REF )
of Conditions C3), C3*), C4'), C6b), C7),
C10), C13), C10*b), and C12*b)
are trivially satisfied by (REF ) and (REF ).
Since inequality (REF ) of Condition C3) is trivially satisfied,
it appears that in Problem C Conditions C1) through C3)
are more efficient than Conditions C1*) through C4*)
(see Corollary REF  (i) and Examples REF 
and REF ).
The following corollary follows from relations
(REF ) through (REF ) and Theorems REF ,
REF  (a), and REF  (b).
Corollary 5.5 
(i) If relations (REF ) of C3) and (REF ) of C4')
are satisfied, then
\(\sup _{f\in B\setminus \lbrace 0\rbrace } E\left(f,G,F^{(1)}_{*}\right)\le \liminf _{n\rightarrow {\infty }}\sup _{f_n\in B_n}E\left(f_n,G_n,F^{(1)}_{*n}\right).\) 
(ii) Let \(\sup _{f_n\in B_n}E\left(f_n,G_n,F^{(1)}_{*n}\right)\)  be a
finite number for every \(n\in {\mathbb {N}}\) .
If Conditions C6*a) and C10*a) are satisfied, then
\(\sup _{f\in B\setminus \lbrace 0\rbrace } E\left(f,G,F^{(1)}_{*}\right)\ge \limsup _{n\rightarrow {\infty }}\sup _{f_n\in B_n}E\left(f_n,G_n,F^{(1)}_{*n}\right).\) 
(iii) Let \(\sup _{f_n\in B_n}E\left(f_n,G_n,F^{(1)}_{*n}\right)\)  be a
finite number for every \(n\in {\mathbb {N}}\) .
If Conditions C6*a) and C12*a) are satisfied, then (REF ) holds true.

Remark 5.6 
Note that relations
\(B\subseteq F^{(1)}\cap F^{(2)}\)  and
\(D(B)\subseteq B\)  of Condition C4') are satisfied because
\(F^{(2)}=F^{(1)}\)  and \(D=I\)  is an imbedding operator by (REF ).

Remark 5.7 
For linear operators \(L_n^{(1)},\,l_n^{(1)}\)  and
linear operators \(l_{n,s}^{(1)}\) 
with
\(l_s^{(1)}=l_{n,s}^{(1)}\,l_n^{(1)},\,n\in {\mathbb {N}},\,s\in {\mathbb {N}},\)  and
\(\sup _{n\in {\mathbb {N}},s\in {\mathbb {N}}}\left\Vert l_{n,s}^{(1)}\right\Vert _{F_{*n}^{(1)}\rightarrow F_{*s}^{(1)}}\le 1\) ,
Corollary REF  with slightly different conditions was obtained
in [30]} for normed
spaces \(F_*^{(1)}\)  and \(F_{*n}^{(1)},\,n\in {\mathbb {N}}\) .
For \({\beta }_2\) -normed spaces
\(F_*^{(1)}\)  and \(F_{*n}^{(1)},\,n\in {\mathbb {N}}\) ,
the corresponding results were proved in
[31]}.

Remark 5.8 
Summarizing the applications of the results from Section 
to Problems A, B, and C presented in this section,
we conclude that Corollary REF  is new, while weaker versions
of Corollaries REF  and REF  were obtained in [30]}, [31]}
(see also Remarks REF , REF  and REF ).


Function Spaces, Function Classes, and Operators
Applications of Corollaries REF , REF , and REF  to
concrete problems of approximation theory (see Section )
require more specified spaces.
Here, we introduce some function spaces, function classes,
and operators and study their certain properties.
Notation
Let \({\mathbb {R}}^m\)  be the Euclidean \(m\) -dimensional space with elements
\(x=(x_1,\ldots ,x_m),\, y=(y_1,\ldots ,y_m),\,t=(t_1,\ldots ,t_m)\) ,
the inner product \(t\cdot y:=\sum _{j=1}^mt_jy_j\) ,
and the norm \(\vert t\vert :=\sqrt{t\cdot t}\) .
Next, \({\mathbb {C}}^m:={\mathbb {R}}^m+i{\mathbb {R}}^m\)  is the \(m\) -dimensional
complex space with elements
\(z=(z_1,\ldots , z_m)=x+iy\) 
and the norm \(\vert z\vert :=\sqrt{\vert x\vert ^2+\vert y\vert ^2}\) ;
\({\mathbb {Z}}^m\)  denotes the set of all integral lattice points
\(k=(k_1,\ldots ,k_m)\)  in \({\mathbb {R}}^m\) ;
and \({\mathbb {Z}}^m_+\)  is a subset of \({\mathbb {Z}}^m\) 
of all points with nonnegative coordinates.
We also use multi-indices \({\beta }=({\beta }_1,\ldots ,{\beta }_m)\in {\mathbb {Z}}^m_+\) 
and \({\alpha }=({\alpha }_1,\ldots ,{\alpha }_m)\in {\mathbb {Z}}^m_+\) 
with
\(\vert {\beta }\vert :=\sum _{j=1}^m{\beta }_j,\qquad \vert {\alpha }\vert :=\sum _{j=1}^m{\alpha }_j,\qquad y^{\beta }:=y_1^{{\beta }_1}\cdot \cdot \cdot y_m^{{\beta }_m}, \qquad D^{\alpha }:=\frac{\partial ^{{\alpha }_1}}{\partial y_1^{{\alpha }_1}}\cdot \cdot \cdot \frac{\partial ^{{\alpha }_m}}{\partial y_m^{{\alpha }_m}}.\) 
Let \(V\)  be a centrally symmetric (with respect to the origin)
closed
convex body in \({\mathbb {R}}^m\)  and
\(V^*:=\lbrace y\in {\mathbb {R}}^m: \forall \, t\in V, \vert t\cdot y\vert \le 1\rbrace \) 
be the polar of \(V\) .
It is well known that \(V^*\)  is a centrally symmetric (with respect to the origin)
closed
convex body in \({\mathbb {R}}^m\)  and \(V^{**} =V\)  (see, e.g., [40]}).
The set \(V\)  generates the following dual norms
on \({\mathbb {R}}^m\)  and \({\mathbb {C}}^m\)  by
\(\Vert y\Vert _V^*:=\sup _{t\in V}\vert t\cdot y\vert ,\quad y\in {\mathbb {R}}^m;\qquad \Vert z\Vert _V^*:=\sup _{t\in V}\left|\sum _{j=1}^m t_jz_j\right|,\quad z\in {\mathbb {C}}^m.\) 
Note also that \(V^*\)  is the unit ball in the norm \(\Vert \cdot \Vert _V^*\)  on \({\mathbb {R}}^m\) .
Examples of sets \(V\) 
and \(V^*\)  and dual norms \(\Vert \cdot \Vert _V^*\)  are given below.
Given \({\sigma }\in {\mathbb {R}}^m,\,{\sigma }_j>0,\,1\le j\le m\) , and \(M>0\) , let
\(\Pi ^m_{\sigma }:=\lbrace t\in {\mathbb {R}}^m: \vert t_j\vert \le {\sigma }_j, 1\le j\le m\rbrace ,\,Q^m_M:=\lbrace t\in {\mathbb {R}}^m: \vert t_j\vert \le M, 1\le j\le m\rbrace \) ,
and
\({\mathfrak {B}}^m_M:=\lbrace t\in {\mathbb {R}}^m: \vert t\vert \le M\rbrace \) 
be the \(m\) -dimensional parallelepiped, cube, and ball, respectively.
For example, if \({\sigma }\in {\mathbb {R}}^m,\,{\sigma }_j>0,\,1\le j\le m\) , and \(V=\left\lbrace t\in {\mathbb {R}}^m:\left(\sum _{j=1}^m\vert t_j/{\sigma }_j\vert ^{\mu }\right)^{1/\mu }\le 1\right\rbrace \) , then
for \(y\in {\mathbb {R}}^m,\,\Vert y\Vert _V^*=\left(\sum _{j=1}^m\vert {\sigma }_j y_j\vert ^{\lambda }\right)^{1/{\lambda }}\) 
and \(V^*=\left\lbrace y\in {\mathbb {R}}^m:\left(\sum _{j=1}^m\vert {\sigma }_j y_j\vert ^{\lambda }\right)^{1/{\lambda }}\le 1\right\rbrace \) ,
where \(\mu \in [1,{\infty }],\,{\lambda }\in [1,{\infty }]\) , and \(1/\mu +1/{\lambda }=1\) .
In particular,
\(\Vert y\Vert _{\Pi ^m_{\sigma }}^*=\sum _{j=1}^m{\sigma }_j\vert y_j\vert ,\,\Vert y\Vert _{Q^m_M}^*=M\sum _{j=1}^m\vert y_j\vert \) ,
and \(\Vert y\Vert _{{\mathfrak {B}}^m_M}^*=M\vert y\vert \) .
We also use the floor function \(\lfloor \cdot \rfloor \) .

Function spaces
In the capacity of spaces \(F^{(1)},\,F^{(2)},\,F^{(1)}_n,\)  and
\(F^{(2)}_n\)  from Sections , , and 
or their \(\ast \) -modifications from Section ,
we mostly use the following function spaces.
Let \(\mu \)  be a \({\sigma }\) -finite positive measure on a \({\sigma }\) -algebra \(\Sigma \)  over \({\mathbb {R}}^m\) 
and let \(S_\mu ({\mathbb {R}}^m)\)  be a vector space of all \(\mu \) -measurable functions
\(f:{\mathbb {R}}^m\rightarrow {\mathbb {C}}^1\) .
Next, let \(F({\mathbb {R}}^m)=F({\mathbb {R}}^m,\mu )\)  be a \({\beta }\) -normed space, \({\beta }\in (0,1]\) ,
of functions from \(S_\mu ({\mathbb {R}}^m)\) .
We assume that the \({\beta }\) -norm of \(F({\mathbb {R}}^m)\)  possesses
properties of monotonicity and \({\omega }\) -continuity.
Definition 6.1 
We say that the \({\beta }\) -norm \(\Vert \cdot \Vert _{F({\mathbb {R}}^m)}\)  is
monotone if relations
\(g\in S_\mu ({\mathbb {R}}^m),\, f\in F({\mathbb {R}}^m)\) ,
and \(\vert g(x)\vert \le \vert f(x)\vert ,\,x\in {\mathbb {R}}^m\) , imply
\(g\in F({\mathbb {R}}^m)\)  and
\(\Vert g\Vert _{F({\mathbb {R}}^m)}\le \Vert f\Vert _{F({\mathbb {R}}^m)}\) .

Given a \(\mu \) -measurable set \({\Omega }\subset {\mathbb {R}}^m\)  and a space
\(F({\mathbb {R}}^m)\) ,
we define a space \(F({\Omega })\)  of the restrictions
\(f\left|_{\Omega }\right.\) 
of functions \(f\in S_\mu ({\mathbb {R}}^m)\)  to \({\Omega }\)  with the finite
\({\beta }\) -norm \(\Vert f\left|_{\Omega }\right.\Vert _{F({\Omega })}:=\Vert f\cdot \chi ({\Omega })\Vert _{F({\mathbb {R}}^m)}\) ,
where \(\chi ({\Omega })\)  is the characteristic function of \({\Omega }\) .
If there is no confusion, we use the notation \(f\) 
instead of \(f\left|_{\Omega }\right.\) .
For a monotone \({\beta }\) -norm \(\Vert \cdot \Vert _{F({\mathbb {R}}^m)}\) ,
\(\Vert f\Vert _{F({\Omega })}\le \Vert f\Vert _{F({\mathbb {R}}^m)},\qquad f\in F({\mathbb {R}}^m),\) 
so if
\(f\in F({\mathbb {R}}^m)\) , then \(f=f\left|_{\Omega }\right.\in F({\Omega })\) .
In a sense, the next definition allows us to reverse
the inequality sign in (REF ).
Definition 6.2 
We say that the monotone \({\beta }\) -norm \(\Vert \cdot \Vert _{F({\mathbb {R}}^m)}\)  is
\({\omega }\) -continuous if for any increasing sequence of \(\mu \) -measurable
sets \({\omega }:=\lbrace {\Omega }_n\rbrace _{n=1}^{\infty }\) 
in \({\mathbb {R}}^m\)  with \(\cup _{n=1}^{\infty }{\Omega }_n={\mathbb {R}}^m\) ,
the following relation is valid:
\(\lim _{n\rightarrow {\infty }}\Vert f\Vert _{F({\Omega }_n)}= \Vert f\Vert _{F({\mathbb {R}}^m)},\qquad f\in F({\mathbb {R}}^m).\) 

Examples of spaces with monotone and \({\omega }\) -continuous norms include certain
Orlicz, Lorentz, and Marcinkiewicz spaces (see, e.g.
[41]}). In particular, normed
separable spaces \(F({\mathbb {R}}^m)\)  or normed spaces,
satisfying the Fatou condition (named C-condition in
[41]}), have \({\omega }\) -continuous norms.
A typical example of a space with a monotone and
\({\omega }\) -continuous \({\beta }\) -norm is
the space \(L_{r,\mu }({\mathbb {R}}^m), r\in (0,{\infty }]\) ,
of all \(\mu \) -measurable complex-valued functions \(f\) 
on \({\mathbb {R}}^m\)  with the finite \({\beta }\) -norm
\(\Vert f\Vert _{L_{r,\mu }({\mathbb {R}}^m)}:=\left\lbrace \begin{array}{ll}\left(\int _{{\mathbb {R}}^m}\vert f(x)\vert ^r d\mu (x)\right)^{1/r}, & 0<r<{\infty },\\\operatornamewithlimits{ess\,sup}_{x\in {\mathbb {R}}^m} \vert f(x)\vert , &r={\infty },\end{array}\right.\) 
where \({\beta }=\min \lbrace 1,r\rbrace \) .
For a \(\mu \) -measurable set \({\Omega }\subset {\mathbb {R}}^m\) , we also use
the \({\beta }\) -normed space \(L_{r,\mu }({\Omega })\)  (\(r\in (0,{\infty }],\,{\beta }=\min \lbrace 1,r\rbrace \) ).
In all examples of Chapter ,
\(\mu \)  is a weighted measure on \({\mathbb {R}}^m\) , i.e.,
\(d\mu (x)=W(x)dx,\,x\in {\mathbb {R}}^m\) , where \(W\)  is a locally Lebesgue-integrable
weight on \({\mathbb {R}}^m\) .
In particular, we use the weighted measure with a univariate
power weight
\(W(x)=\vert x\vert ^{2\nu +1}\)  in Example REF .
In case of the Lebesgue measure \(\mu \)  (i.e, \(W=1\) ), we use the notation
\(L_{r}({\mathbb {R}}^m)\)  and \(L_{r}({\Omega }),\,r\in (0,{\infty }]\) , in Examples
REF , REF , REF , REF ,
REF , and REF .
In Example REF  we use a univariate exponential weight
\(W(x)=W_r(x):=\exp (-rQ(x)),\,r\in (0,{\infty })\) ,
where \(Q:{\mathbb {R}}^1\rightarrow [0,{\infty })\)  is an even differentiable
convex function (see, e.g., [43]}, [44]}
for applications of \(W_r\)  in analysis).
A typical example of \(W_r\)  is
a Freud weight \(W_{r,{\alpha }}(x):=\exp (-r\vert x\vert ^{\alpha })),\,{\alpha }>1\) .
The corresponding \({\beta }\) -norm is a version of (REF )
in the following form:
\(\Vert f\Vert _{L_{r}(W_r,{\mathbb {R}}^1)}:=\left(\int _{{\mathbb {R}}^1}\vert f(x)\exp (-Q(x))\vert ^r dx\right)^{1/r}, \qquad 0<r<{\infty },\) 
where \({\beta }=\min \lbrace 1,r\rbrace \) .
An extension of (REF ) to \(r={\infty }\)  is given by
\(\Vert f\Vert _{L_{{\infty }}(W_{\infty },{\mathbb {R}}^1)}:=\operatornamewithlimits{ess\,sup}_{x\in R^1} \vert f(x)\exp (-Q(x))\vert .\) 
The space \(L_{r}(W_r,{\mathbb {R}}^1)\)  (\(r\in (0,{\infty }],\,{\beta }=\min \lbrace 1,r\rbrace \) ) is the space
of all Lebesgue-measurable complex-valued functions \(f\) 
on \({\mathbb {R}}^1\)  with the finite
monotone and
\({\omega }\) -continuous \({\beta }\) -norms (REF ) and (REF ).
Note that norm (REF ) is different, compared with
norm (REF ) for \(r={\infty }\) .
One more typical example of a space
with a monotone and
\({\omega }\) -continuous norm is
the normed space \(C_0({\mathbb {R}}^m)\)  of all complex-valued
continuous functions \(f\)  on \({\mathbb {R}}^m\)  with the "point" norm
\(\Vert f\Vert _{C_0({\mathbb {R}}^m)}:=\vert f(0)\vert \) .
For a compact set \({\Omega }\subset {\mathbb {R}}^m,\,0\in {\Omega },\) 
we can also define the normed space \(C_0({\Omega })\) .
We use these norms in Examples REF  and REF .
Given \(a>0\) , the space of all functions \(f\)  from \(L_{\infty }({\mathbb {R}}^m)\) 
that are \(2\pi a\) -periodic in each variable is denoted by
\(\tilde{L}_{{\infty },2\pi a,m}\)  with the norm
\(\Vert f\Vert _{\tilde{L}_{{\infty },2\pi a,m}}=\Vert f\Vert _{L_{\infty }(R^m)}=\Vert f\Vert _{L_{\infty }(Q^m_{\pi a})}\)  (see Example REF ).

Function classes
In the capacity of sets \(B\)  and
\(B_n\)  from Sections , , and ,
we use the following function classes.
Let \({\mathcal {P}}_{n,m}\)  be a set of all
polynomials \(P(y)=\sum _{\vert {\beta }\vert \le n}c_{\beta }y^{\beta }\) 
in \(m\)  variables of degree at most \(n,\,n\in Z^1_+\) ,
with complex coefficients.
Given \(a>0\) , the set of all trigonometric polynomials
\(Q(x)=\sum _{k\in aV\cap {\mathbb {Z}}^m}c_ke^{(ik)\cdot x}\)  with complex
coefficients is denoted by \({\mathcal {T}}_{aV}\) 
and the set of all trigonometric polynomials
\(T(x)=\sum _{k\in aV\cap {\mathbb {Z}}^m}c_ke^{(ik)\cdot (x/a)}\)  with complex
coefficients is denoted by \({\mathcal {T}}_{a,V}\) .
Definition 6.3 
We say that an entire function \(f:{\mathbb {C}}^m\rightarrow {\mathbb {C}}^1\)  has exponential type \(V\) 
if for any \({\varepsilon }>0\)  there exists a constant \(C_0({\varepsilon },f)\)  such that
for all \(z\in {\mathbb {C}}\) ,
\(\vert f(z)\vert \le C_0({\varepsilon },f)\exp \left((1+{\varepsilon })\Vert z\Vert _V^*\right)\) .

The set of all entire function of exponential type \(V\)  is denoted
by \(B_V\) .
The set \(B_V\)  was defined by Stein and Weiss
[45]}.
For \(V=\Pi ^m_{\sigma },\,V=Q^m_M,\)  and \(V={\mathfrak {B}}^m_M\) ,
similar
sets of entire functions were
defined by Bernstein [46]} and Nikolskii
[47]}, [24]}, see also
[10]}.
Properties of functions from \(B_V\)  have been investigated in numerous
publications (see, e.g., [46]}, [47]}, [24]}, [45]}, [54]}, [55]}, [56]}, [7]} and
references therein).
Throughout the remaining sections of the paper,
if no confusion may occur, the same notation
is applied to functions
\(f\in B_V,\,f\in {\mathcal {T}}_{aV},\,f\in {\mathcal {T}}_{a,V}\) , and \(f\in {\mathcal {P}}_{n,m}\) 
and their restrictions to \({\mathbb {R}}^m\) 
or to a subset \({\Omega }\)  of \({\mathbb {R}}^m\)  (e.g., in the form
\(f\in B_V\cap L_p({\mathbb {R}}^m))\) .
Let \(M_{C,N}\)  be the set of all Lebesgue measurable functions \(f\) ,
satisfying the inequality
\(\vert f(x)\vert \le C(1+\vert x\vert )^N\) 
for a. a. \(x\in {\mathbb {R}}^m\) ,
where \(C>0\)  and \(N\ge 0\)  are constants independent of \(x\) .
In particular, we discuss classes \({\mathcal {T}}_{a,V}\cap M_{C,N}\) 
in Examples REF  and REF .
In addition, let \({\omega }:[0,{\infty })\rightarrow [0,{\infty })\)  be a modulus of continuity,
i.e., \({\omega }(0)=0\)  and \({\omega }\)  is a continuous, nondecreasing, and
subadditive function on \([0,{\infty })\) .
Next, we define the class \(H_{\omega }(\Omega )\) 
of all functions \(f\)  on \(\Omega \subseteq {\mathbb {R}}^m\) ,
satisfying the inequality
\(\vert f(x)-f(y)\vert \le {\omega }(\vert x-y\vert ),\qquad x\in \Omega ,\quad y\in \Omega .\) 
We also need a subclass
\(H_{\omega ,0}({\mathbb {R}}^m):=\lbrace f\in H_{\omega }({\mathbb {R}}^m):f(0)=0\rbrace \)  of \(H_{\omega }({\mathbb {R}}^m)\) .
Note that
\(H_{\omega ,0}({\mathbb {R}}^m)\subseteq M_{{\omega }(2),1}.\) 
Indeed, if \(f\in H_{\omega ,0}({\mathbb {R}}^m)\) , then
\(\vert f(x)\vert \le {\omega }(\vert x\vert ),\,x\in {\mathbb {R}}^m\) . Next,
\(\max _{\tau \in [0,2]}{\omega }(\tau )={\omega }(2)\) , and
if \(2^k<\tau \le 2^{k+1},\,k\in {\mathbb {N}}\) ,
then \({\omega }(\tau )\le {\omega }(2^{k+1})< \tau {\omega }(2)\) . Therefore,
\(\vert f(x)\vert \le {\omega }(2)(1+\vert x\vert ),\, x\in {\mathbb {R}}^m\) ,
and (REF ) holds true.
In case of \({\omega }(\tau )=\tau ^{\lambda },\,\tau \in [0,{\infty }),\, 0<{\lambda }\le 1\) , the class
\(H^{\lambda }(\Omega ):=H_{\omega }(\Omega )\)  is called the Hölder class.
For \(m=1\)  we also define the class \(W^sH_{\omega }({\mathbb {R}}^1)\) 
of all \(s\) -differentiable functions on \({\mathbb {R}}^1\)  with
\(f^{(s)}\in H_{\omega }({\mathbb {R}}^1),\,s\in {\mathbb {Z}}^1_+\) .
We also recall that the error of best approximation of
\({\varphi }\in F({\Omega })\)  by elements from a subspace \(B\)  of \(F({\Omega })\) 
is defined by
\(E({\varphi },B,F({\Omega })):=\inf _{f\in B}\Vert {\varphi }-f\Vert _{F({\Omega })}\) 
(see (REF )).
Certain properties of functions from defined classes
are discussed below. We first prove multivariate versions
of two known univariate results
(see, e.g., [1]}).
Lemma 6.4 
If a function \(f\in B_V\)  has a period of \(2\pi a\)  in each variable,
then \(f\in {\mathcal {T}}_{a,V},\,a>0\) .

The entire function \(f\)  coincides with its Fourier series
\(\sum _{k\in {\mathbb {Z}}^m}c_ke^{(ik)\cdot (x/a)}\) .
Since \(f\in L_{\infty }({\mathbb {R}}^m)\) ,
this function is a tempered distribution on \({\mathbb {R}}^m\) .
Then by the generalized Paley-Wiener theorem [54]},
the support of the Fourier transform of \(f\)  is a subset of \(V\) .
So \(c_k=0\)  for \(k\in {\mathbb {Z}}^m\setminus aV\) . \(\Box \)
Lemma 6.5 
If \({\varphi }\in \tilde{L}_{{\infty },2\pi a,m}\) , then
\(E({\varphi },B_V,L_{\infty }({\mathbb {R}}^m))=E({\varphi },{\mathcal {T}}_{a,V},L_{\infty }({\mathbb {R}}^m)).\) 

Since \({\mathcal {T}}_{a,V}\subseteq B_V\) , we have
\(E({\varphi },B_V,L_{\infty }({\mathbb {R}}^m))\le E({\varphi },{\mathcal {T}}_{a,V},L_{\infty }({\mathbb {R}}^m)).\) 
Next given \({\varepsilon }\in (0,{\infty })\) , let \(g_{\varepsilon }\in B_V\cap L_{\infty }({\mathbb {R}}^m)\) 
satisfy the relation
\(\Vert {\varphi }-g_{\varepsilon }\Vert _{L_{\infty }({\mathbb {R}}^m)}<E({\varphi },B_V,L_{\infty }({\mathbb {R}}^m))+{\varepsilon }.\) 
Then the function
\(G_N(x):=(2N+1)^{-m}\sum _{k\in Q_N^m}g_{\varepsilon }(x+2\pi a k),\qquad N\in {\mathbb {N}},\) 
belongs to \(B_V\)  and by (REF ),
\(\Vert {\varphi }-G_N\Vert _{L_{\infty }({\mathbb {R}}^m)}<E({\varphi },B_V,L_{\infty }({\mathbb {R}}^m))+{\varepsilon }.\) 
Moreover,
\(\sup _{N\in {\mathbb {N}}}\Vert G_N\Vert _{L_{\infty }({\mathbb {R}}^m)}\le \Vert g_{\varepsilon }\Vert _{L_{\infty }({\mathbb {R}}^m)}\) .
Applying the compactness theorem
(see [15]}) to the sequence
\(\lbrace G_N\rbrace _{N=1}^{\infty }\)  we see that there exist a sequence
of natural numbers \(\lbrace N_s\rbrace _{s=1}^{\infty }\)  and a function
\(G_0\in B_V\)  such that
\(\lim _{s\rightarrow {\infty }} G_{N_s}(x)=G_0(x)\) 
uniformly on any compact \(K\subset {\mathbb {R}}^m\) . Then
(REF ) and (REF ) imply that
\(\Vert {\varphi }-G_0\Vert _{L_{\infty }({\mathbb {R}}^m)}\le E({\varphi },B_V,L_{\infty }({\mathbb {R}}^m))+{\varepsilon }.\) 
Next, denoting by \(e_j\)  the \(j\) th unit vector
in \({\mathbb {R}}^m,\,1\le j\le m\) ,
we have
\(\max _{1\le j\le m}\left\Vert G_N(\cdot +2\pi a e_j)-G_N(\cdot )\right\Vert _{L_{\infty }(R^m)}\le C_7/N,\) 
where a constant \(C_7>0\)  is independent of \(N\) . Setting \(N=N_s\)  and
letting \(s\rightarrow {\infty }\)  in (REF ),
we see from (REF ) and (REF )
that \(G_0\)  is \(2\pi a\) -periodic
in each variable.
Therefore, \(G_0\in {\mathcal {T}}_{a,V}\)  by Lemma REF ,
and it follows from (REF ) that
\(E({\varphi },{\mathcal {T}}_{a,V},L_{\infty }({\mathbb {R}}^m))\le \Vert {\varphi }-G_0\Vert _{L_{\infty }({\mathbb {R}}^m)}\le E({\varphi },B_V, L_{\infty }({\mathbb {R}}^m))+{\varepsilon }.\) 
Letting \({\varepsilon }\rightarrow 0+\)  in (REF ) and taking account of
(REF ), we arrive at (REF ).\(\Box \)

We also need the following extension theorem.
Lemma 6.6 
Given function \(f:Q_M^m\rightarrow {\mathbb {R}}^1,\,M>0\) , there exists a function
\(\tilde{f}:{\mathbb {R}}^m\rightarrow {\mathbb {R}}^1\)  such that \(\tilde{f}=f\)  on \(Q_M^m\)  and
\(\tilde{f}\)  is \(4M\) -periodic in each variable. In addition,
if \(f\in H_{\omega }(Q_M^m)\) , then \(\tilde{f}\in H_{\omega }({\mathbb {R}}^m)\) .

Let us set
\(h_p(x):=(-1)^{k_p}(x_p-2Mk_p),\quad x\in Q_M^m+2Mk, \quad 1\le p\le m,\quad k=(k_1,\ldots , k_m)\in {\mathbb {Z}}^m.\) 
Then \(h(x)=(h_1(x),\ldots ,h_m(x)):{\mathbb {R}}^m\rightarrow Q_M^m\) 
is a vector-valued function. Indeed, this statement is obviously
true for any interior point \(x\)  of \(Q_M^m+2Mk,\,k\in {\mathbb {Z}}^m\) .
Let \(x\)  belong to the boundary of \(Q_M^m+2Mk,\,k\in {\mathbb {Z}}^m\) .
Then without loss of generality we can assume that
there exists \(s\in {\mathbb {N}},\,1\le s\le m\) , such that
\(&&x_p=-M+2Mk_p=M+2M(k_p-1),\qquad 1\le p\le s,\\&&x_p=x_p^*+2Mk_p,\qquad \vert x_p^*\vert <M,\qquad s+1\le p\le m.\) 
It follows from (REF ) that \(h(x_p)=(-1)^{k_p+1}M,\,1\le p\le s\) , and \(h(x_p)\) 
for \(s+1\le p\le m\)  is uniquely defined by (REF )
as well. Therefore, \(h\)  is a function.
Next, \(h\)  is \(4M\) -periodic in each variable.
Indeed, if \(x\in Q_M^m+2Mk,\,k\in {\mathbb {Z}}^m\) , then for every
\(l=(l_1,\ldots , l_m) \in {\mathbb {Z}}^m\) ,
\(h_p(x+4Ml)=(-1)^{k_p+2l_p} (x_p+4Ml_p-2M(k_p+2l_p))=h_p(x),\qquad 1\le p\le m.\) 
In addition, \(h(x)=x\)  on \(Q_M^m\) . Then
\(\vert h(x)-h(x^*)\vert \le \vert x-x^*\vert ,\,x\in {\mathbb {R}}^m,\,x^*\in {\mathbb {R}}^m.\) 
Indeed, it suffices to show that \(y_p=h_p(x_p)\in H^1({\mathbb {R}}^1),\,1\le p\le m\) .
Since \(y_p\)  is \(4M\) -periodic on \({\mathbb {R}}^1\)  and the curve has
the slope of 1 on \([-M,M]\)  and the slope of \(-1\)  on
\([-2M,-M]\)  and \([M,2M]\) , we see that \(y_p\in H^1({\mathbb {R}}^1),\,1\le p\le m\) .
Finally, the function \(\tilde{f}(x):=f(h(x))\)  satisfies
all conditions of the lemma.\(\Box \)

Note that for \(m=1\)  this result can be found in
[1]},
while for \(m>1\)  Shvedov [62]}
presented a geometric construction of \(\tilde{f}\) 
with no proof provided.
Operators
In the capacity of operators \(D\)  and \(D_n\)  from Sections ,
, and ,
we use linear operators, in particular, linear
differential operators of the form
\(d_N:=\sum _{\vert {\alpha }\vert =N}b_{\alpha }D^{\alpha }\) 
with constant coefficients \(b_{\alpha }\in {\mathbb {C}}^1,\,{\alpha }\in {\mathbb {Z}}^m_+,\,\vert {\alpha }\vert =N,\,N\in {\mathbb {Z}}^1_+\) .
In case of \(N=0\)  we assume that \(d_N\) 
is the corresponding imbedding (or identity) operator \(I\) .
In the capacity of operators \(L_n^{(1)},\,L_n^{(2)},\,l_n^{(1)},\,l_n^{(2)},\,l_{n,s}^{(1)}\)  and \(l_{n,s}^{(2)}\) 
from the conditions of Section REF 
and also from Sections  and , we use the
following linear restriction operator.
Let \({\Omega }_1\)  and \({\Omega }_2\)  be \(\mu \) -measurable subsets of \({\mathbb {R}}^m\)  such that
\({\Omega }_2\subseteq {\Omega }_1\)  and let
\(R=R({\Omega }_1,{\Omega }_2):F({\Omega }_1)\rightarrow F({\Omega }_2)\)  be the restriction operator.
For \(\Omega \subseteq {\mathbb {R}}^m\)  we also use the notation
\(D_n\left|_{\Omega }\right.:=D_nR({\mathbb {R}}^m,\Omega )\) .
If no confusion may occur, we often use the notation \(D_n\) 
instead of \(D_n\left|_{\Omega }\right.\) .
The following proposition is a direct consequence of (REF ) and
(REF ).
Proposition 6.7 
(i) If \(F({\mathbb {R}}^m)\)  is a space with a monotone norm, then
\(\Vert R({\Omega }_1,{\Omega }_2)\Vert \le 1\) , i.e., for any \(h\in F({\Omega }_1)\) ,
\(\Vert R({\Omega }_1,{\Omega }_2)\,(h)\Vert _{F({\Omega }_2)}\le \Vert h\Vert _{F({\Omega }_1)}.\) 
(ii) Let \({\omega }=\lbrace {\Omega }_n\rbrace _{n=1}^{\infty }\)  be
any increasing sequence of \(\mu \) -measurable
sets
in \({\mathbb {R}}^m\)  with \(\cup _{n=1}^{\infty }{\Omega }_n={\mathbb {R}}^m\) .
If \(F({\mathbb {R}}^m)\)  is a space with a \({\omega }\) -continuous norm, then
for any \(h\in F({\mathbb {R}}^m)\) ,
\(\lim _{n\rightarrow {\infty }}\Vert R({\mathbb {R}}^m,{\Omega }_n)\,(h)\Vert _{F({\Omega }_n)}= \Vert h\Vert _{F({\mathbb {R}}^m)}.\) 

In addition, for a set \(A\)  of functions
\(f: {\mathbb {R}}^m \rightarrow {\mathbb {C}}^1\)  we define by
\(A\left|_\Omega \right.:=\lbrace R({\mathbb {R}}^m,\Omega )(f):f\in A\rbrace \) 
the set of the restrictions to \(\Omega \subseteq {\mathbb {R}}^m\) 
of functions from \(A\) .
In Section , if no confusion may occur,
we often do not use this restriction
notation for sets
\(B_V,\,{\mathcal {T}}_{aV},\,{\mathcal {T}}_{a,V},\,{\mathcal {P}}_{n,m}\) ,
and their subsets.

Examples of Asymptotic Relations between Sharp
Constants in Function Spaces
In this section we discuss specified conditions and examples of
relations between sharp constants
in function spaces separately for each major Problem A, B, and C
of approximation theory described in Sections  and .
Throughout the section \(\lbrace {\Omega }_n\rbrace _{n=1}^{\infty }\)  and
\(\lbrace {\Omega }_n^*\rbrace _{n=1}^{\infty }\)  are
increasing sequences of \(\mu \) -measurable
sets
in \({\mathbb {R}}^m\)  with \(\cup _{n=1}^{\infty }{\Omega }_n={\mathbb {R}}^m\) 
and \(\cup _{n=1}^{\infty }{\Omega }_n^*={\mathbb {R}}^m\) .
Inequalities for Approximating Elements
(Problem A)
Here, we discuss a special case of Corollary REF  and
three examples.

General result.
We assume that all assumptions of Section REF  are held
in the following special cases.
First,
we assume that \(F^{(j)}=F^{(j)}({\mathbb {R}}^m)=F^{(j)}({\mathbb {R}}^m,\mu )\)  is
a \({\beta }_j\) -normed functional space
with the monotone and \({\omega }\) -continuous
\({\beta }_j\) -norm
(see Definitions REF  and REF ),
\({\beta }_j\in (0,1]\) , and
\(F^{(j)}_n=F^{(j)}({\Omega }_n)\) 
or \(F^{(j)}_n=F^{(j)}({\Omega }_n^*),\, j=1,2,\,n\in {\mathbb {N}}\) .
Next, we define the restriction operators \(L_n^{(1)}=R({\mathbb {R}}^m,{\Omega }_n^*),\,l_n^{(1)}=l_n^{(2)}=R({\mathbb {R}}^m,{\Omega }_n),\,n\in {\mathbb {N}}\) , and
\(l_{n,s}^{(1)}=l_{n,s}^{(2)}=\left\lbrace \begin{array}{ll}R({\Omega }_n,{\Omega }_s), &n\ge s,\\R({\Omega }_s,{\Omega }_n), &n< s,\end{array}\right.\qquad n\in {\mathbb {N}},\,s\in {\mathbb {N}}.\) 
We also assume that \(B,\,B_n\) , and \(B_n^*\) 
are nontrivial subspaces of \(F^{(2)}({\mathbb {R}}^m),\,F^{(2)}({\Omega }_n)\) , and \(F^{(2)}({\Omega }_n^*)\) ,
respectively, \(n\in {\mathbb {N}}\) .
In addition, let \(D:B\rightarrow F^{(1)}({\mathbb {R}}^m),\,D_n:B_n\rightarrow F^{(1)}({\Omega }_n)\) ,
and \(D_n^*:B_n^*\rightarrow F^{(1)}({\Omega }_n^*),\,n\in {\mathbb {N}}\) ,
be linear operators.
Finally, let
\(B_n^\prime \subseteq \left\lbrace f_n\in B_n\setminus \lbrace 0\rbrace :\Vert D_n(f_n)\Vert _{ F^{(1)}({\Omega }_n)}=1\right\rbrace \)  be a set
from Condition C9) such that
the following relations hold:
\(\sup _{f_n\in B_n^\prime }\frac{1}{\Vert f_n\Vert _{F^{(2)}({\Omega }_n)}}=\sup _{f_n\in B_n^\prime }\frac{\Vert D_n(f_n)\Vert _{F^{(1)}({\Omega }_n)}}{\Vert f_n\Vert _{F^{(2)}({\Omega }_n)}}=\sup _{f\in B_n\setminus \lbrace 0\rbrace }\frac{\Vert D_n(f)\Vert _{F^{(1)}({\Omega }_n)}}{\Vert f\Vert _{F^{(2)}({\Omega }_n)}},\qquad n\in {\mathbb {N}}.\) 
The existence of \(B_n^\prime ,\,n\in {\mathbb {N}}\) , follows immediately
from Proposition REF  (i).
Below we introduce some conditions on families
\(\left\lbrace D,B,F^{(1)}({\mathbb {R}}^m),F^{(2)}({\mathbb {R}}^m)\right\rbrace ,\linebreak \left\lbrace D_n,B_n,F^{(1)}({\Omega }_n),F^{(2)}({\Omega }_n)\right\rbrace \) ,
and \(\left\lbrace D_n^*,B_n^*,F^{(1)}({\Omega }_n^*),F^{(2)}({\Omega }_n^*)\right\rbrace ,\,n\in {\mathbb {N}}\) ,
that are needed for the validity of relations like (REF )
and/or (REF ) in function spaces.

Condition C3.1). For every \(f\in B\setminus \lbrace 0\rbrace \) 
there exists a sequence \(f_n^*\in B_n^*,\,n\in {\mathbb {N}}\) ,
such that
\(\limsup _{n\rightarrow {\infty }}\Vert f_n^*\Vert _{F^{(2)}({\Omega }_n^*)}\le \Vert f\Vert _{F^{(2)}({\mathbb {R}}^m)}\) 
and
\(\lim _{n\rightarrow {\infty }}\left\Vert D(f)-D_n^*(f_n^*)\right\Vert _{F^{(1)}({\Omega }_n^*)}=0.\) 
Condition C3.1.1). Let \({\Omega }_n^*\subseteq {\Omega }_n,\,n\in {\mathbb {N}}\) .
For every \(f\in B\setminus \lbrace 0\rbrace \) 
there exists a sequence \(f_n\in B_n,\,n\in {\mathbb {N}}\) ,
such that
\(\limsup _{n\rightarrow {\infty }}\Vert f_n\Vert _{F^{(2)}({\Omega }_n)}\le \Vert f\Vert _{F^{(2)}({\mathbb {R}}^m)}\) 
and
\(\lim _{n\rightarrow {\infty }}\left\Vert D(f)-D_n(f_n)\right\Vert _{F^{(1)}({\Omega }_n^*)}=0.\) 
Condition C10.1). (Compactness condition)
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty }\)  such that
\(f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) ,
there exist a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural numbers and an element \(f\in B\) 
such that
\(&&\lim _{k\rightarrow {\infty }}\left\Vert D_{n_k}(f_{n_k})-D(f)\right\Vert _{F^{(1)}\left({\Omega }_{n_k}\right)}=0,\\&&\lim _{k\rightarrow {\infty }}\left\Vert f_{n_k}-f\right\Vert _{F^{(2)}\left({\Omega }_{n_k}\right)}=0.\) 
Condition C12.1). (Weak compactness condition)
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty }\) 
such that
\(f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) ,
there exists \(f\in B\)  such that
for any fixed \(s\in {\mathbb {N}}\) ,
there exists a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural numbers
(\(n_k=n_k(s),\,k\in {\mathbb {N}}\) )
such that
\(&&\lim _{k\rightarrow {\infty }}\left\Vert D_{n_k}(f_{n_k})-D(f)\right\Vert _{F^{(1)}\left({\Omega }_{s}\right)}=0,\\&&\lim _{k\rightarrow {\infty }}\left\Vert f_{n_k}-f\right\Vert _{F^{(2)}\left({\Omega }_{s}\right)}=0,\) 
and, in addition,
\(\left\Vert D_{n_k}(f_{n_k})\right\Vert _{F^{(1)}\left({\Omega }_{n_k}\right)}=\left\Vert D_{n_k}(f_{n_k})\right\Vert _{F^{(1)}\left({\Omega }_{s}\right)},\quad k\in {\mathbb {N}}.\) 
The following corollary holds true.
Corollary 7.1 
(i) If Condition C3.1)
is satisfied, then
\(\sup _{f\in B\setminus \lbrace 0\rbrace }\frac{\Vert D(f)\Vert _{F^{(1)}({\mathbb {R}}^m)}}{\Vert f\Vert _{F^{(2)}({\mathbb {R}}^m)}}\le \liminf _{n\rightarrow {\infty }}\sup _{f\in B_n^*\setminus \lbrace 0\rbrace }\frac{\Vert D_n^*(f)\Vert _{F^{(1)}({\Omega }_n^*)}}{\Vert f\Vert _{F^{(2)}({\Omega }_n^*)}}.\) 
(ii) If Condition C3.1.1)
is satisfied, then
\(\sup _{f\in B\setminus \lbrace 0\rbrace }\frac{\Vert D(f)\Vert _{F^{(1)}({\mathbb {R}}^m)}}{\Vert f\Vert _{F^{(2)}({\mathbb {R}}^m)}}\le \liminf _{n\rightarrow {\infty }}\sup _{f\in B_n\setminus \lbrace 0\rbrace }\frac{\Vert D_n(f)\Vert _{F^{(1)}({\Omega }_n)}}{\Vert f\Vert _{F^{(2)}({\Omega }_n)}}.\) 
(iii) Let
\((D_n,B_n,F^{(1)}({\Omega }_n),F^{(2)}({\Omega }_n)=\sup _{f\in B_n\setminus \lbrace 0\rbrace }\frac{\Vert D_n(f)\Vert _{F^{(1)}({\Omega }_n)}}{\Vert f\Vert _{F^{(2)}({\Omega }_n)}}\) 
(see ())
be a finite number for every \(n\in {\mathbb {N}}\) , and let
Condition C10.1) be satisfied. Then
\(\sup _{f\in B\setminus \lbrace 0\rbrace }\frac{\Vert D(f)\Vert _{F^{(1)}({\mathbb {R}}^m)}}{\Vert f\Vert _{F^{(2)}({\mathbb {R}}^m)}}\ge \limsup _{n\rightarrow {\infty }}\sup _{f\in B_n\setminus \lbrace 0\rbrace }\frac{\Vert D_n(f)\Vert _{F^{(1)}({\Omega }_n)}}{\Vert f\Vert _{F^{(2)}({\Omega }_n)}}.\) 
(iv) Let \((D_n,B_n,F^{(1)}({\Omega }_n),F^{(2)}({\Omega }_n)\) 
be a finite number for every \(n\in {\mathbb {N}}\) , and let
Condition C12.1) be satisfied.
Then (REF ) holds
true.
(v) Let \({\Omega }_n^*\subseteq {\Omega }_n\)  and let \(B_n^*\)  be a
set of the restrictions \((f_n)^*:=R({\mathbb {R}}^m,\Omega _n^*)(f_n)\) 
to \({\Omega }_n^*\) 
of all elements \(f_n\)  from \(B_n,\,n\in {\mathbb {N}}\) .
In addition, let
\(D_n^*((f_n)^*):=(D_n(f_n))^*,\,n\in {\mathbb {N}}\) .
Next,
let \((D_n,B_n,F^{(1)}({\Omega }_n),F^{(2)}({\Omega }_n)\) 
be a finite number for every \(n\in {\mathbb {N}}\) , and let
Condition C3.1) and either C10.1) or C12.1) be satisfied.
Then
\(\limsup _{n\rightarrow {\infty }}\sup _{f\in B_n\setminus \lbrace 0\rbrace }\frac{\Vert D_n(f)\Vert _{F^{(1)}({\Omega }_n)}}{\Vert f\Vert _{F^{(2)}({\Omega }_n)}}&\le & \sup _{f\in B\setminus \lbrace 0\rbrace }\frac{\Vert D(f)\Vert _{F^{(1)}({\mathbb {R}}^m)}}{\Vert f\Vert _{F^{(2)}({\mathbb {R}}^m)}}\nonumber \\&\le &\liminf _{n\rightarrow {\infty }}\sup _{f\in B_n^*\setminus \lbrace 0\rbrace }\frac{\Vert D_n^*(f)\Vert _{F^{(1)}({\Omega }_n^*)}}{\Vert f\Vert _{F^{(2)}({\Omega }_n^*)}}.\) 
(vi) Let
\((D_n,B_n,F^{(1)}({\Omega }_n),F^{(2)}({\Omega }_n)\) 
be a finite number for every \(n\in {\mathbb {N}}\) , and let
Condition C3.1.1) and either C10.1) or C12.1) be satisfied.
Then
\(\sup _{f\in B\setminus \lbrace 0\rbrace }\frac{\Vert D(f)\Vert _{F^{(1)}({\mathbb {R}}^m)}}{\Vert f\Vert _{F^{(2)}({\mathbb {R}}^m)}}= \lim _{n\rightarrow {\infty }}\sup _{f\in B_n\setminus \lbrace 0\rbrace }\frac{\Vert D_n(f)\Vert _{F^{(1)}({\Omega }_n)}}{\Vert f\Vert _{F^{(2)}({\Omega }_n)}}.\) 

Note first that in this proof all elements in discussed
Conditions C1) through C13) are replaced without further notice
by the corresponding elements that are used in Conditions
C3.1), C3.1.1), C10.1), C12.1) and relations (REF )
through (REF ). For example, in Condition
C3) \(B_n\)  is replaced by \(B_n^*\) .
By the assumptions about the functional spaces and operators
introduced above, Conditions C2), C4), C6), C6a),
C6b), C11), and C11a)
of Section REF  are satisfied. Indeed,
Conditions C2), C6), and C11) are satisfied by the direct
definitions of the operators whose existence is required
in these conditions.
In addition,
Conditions C4), C6a), C6b), and C11a) are satisfied
by Proposition REF .
Next, Conditions C3.1), C10.1), and C12.1) of this section
are equivalent to Conditions C3), C10), and C12) of Section
REF , respectively.
Therefore, statements (i), (iii), and (iv) of Corollary REF 
follow immediately
from statements (i), (ii), and (iii) of
Corollary REF , respectively.
To prove statement (ii), we recall that
the norm \(\Vert \cdot \Vert _{F^{(1)}({\mathbb {R}}^m)}\)  is monotone and
\({\omega }\) -continuous. So for
any \(f\in B\setminus \lbrace 0\rbrace \) , we obtain by
Proposition REF  and relation (REF )
of Condition C3.1.1)
\(&&\Vert D(f)\Vert _{F^{(1)}({\mathbb {R}}^m)}=\lim _{n\rightarrow {\infty }}\Vert D(f)\Vert _{F^{(1)}({\Omega }_n^*)}\nonumber \\&&\le \liminf _{n\rightarrow {\infty }}\left(\Vert D(f)-D_n(f_n)\Vert _{F^{(1)}({\Omega }_n^*)}^{{\beta }_1}+\Vert D_n(f_n)\Vert _{F^{(1)}({\Omega }_n^*)}^{{\beta }_1}\right)^{1/{\beta }_1}\nonumber \\&&=\liminf _{n\rightarrow {\infty }}\left\Vert D_n(f_n)\right\Vert _{F^{(1)}({\Omega }_n^*)}\le \liminf _{n\rightarrow {\infty }}\left\Vert D_n(f_n)\right\Vert _{F^{(1)}({\Omega }_n)}\nonumber \\&&\le \liminf _{n\rightarrow {\infty }}\sup _{f\in B_n\setminus \lbrace 0\rbrace }\frac{\Vert D_n(f)\Vert _{F^{(1)}({\Omega }_n)}}{\Vert f\Vert _{F^{(2)}({\Omega }_n)}}\limsup _{n\rightarrow {\infty }}\left(\Vert f_n\Vert _{F^{(2)}({\Omega }_n)} \right).\) 
Then (REF ) follows from (REF ) and (REF ).
Finally, statement (v) follows from (i), (iii), and (iv),
and statement (vi) follows from (ii), (iii), and (iv).
\(\Box \)

Examples.
There is a number of publications
[63]}, [64]}, [9]}, [66]}, [67]}, [68]}, [14]}, [15]}, [17]}, [18]} about special cases of Corollary
REF ;
here, we call them Bernstein-Nikolskii type
inequalities or inequalities of different metrics
(see also Section  A).
Examples REF , REF , and REF  below
discuss inequalities (REF ),
(REF ) and (REF )
proved in these papers in various situations.
Example 7.2 
(Multivariate Bernstein-Nikolskii type
inequalities).
Let
\(F^{(1)}({\mathbb {R}}^m)=L_q({\mathbb {R}}^m),\,{\beta }_1=\min \lbrace 1,q\rbrace ,\,F^{(2)}({\mathbb {R}}^m)=L_p({\mathbb {R}}^m),\,{\beta }_2=\min \lbrace 1,p\rbrace ,\,B=B_V\cap L_p({\mathbb {R}}^m),\,0<p\le q\le {\infty },\,{\Omega }_n=Q_{\pi n}^m,\,{\Omega }_n^*=Q_{n^{\delta }}^m,\, {\delta }={\delta }(p,m)\in (0,1),\,B_n={\mathcal {T}}_{n,V}\left|_{\Omega _n}\right.={\mathcal {T}}_{n,V},\,B_n^*={\mathcal {T}}_{n,V}\left|_{\Omega _n^*}\right.={\mathcal {T}}_{n,V},\,n\in {\mathbb {N}}.\) 
Next, let
\(d_N=\sum _{\vert {\alpha }\vert =N}b_{\alpha }D^{\alpha }\) ,
where
\(b_{\alpha }\in {\mathbb {C}}^1\)  (\({\alpha }\in {\mathbb {Z}}^m_+,\,\vert {\alpha }\vert =N\) ), and \(N\in {\mathbb {Z}}^1_+\) ,
are independent of
\(n\) , and let
\(D=d_N\left|_{{\mathbb {R}}^m}\right.=d_N,\,D_n=d_N\left|_{\Omega _n}\right.=d_N,\,D_n^*=d_N\left|_{\Omega _n^*}\right.=d_N,\,n\in {\mathbb {N}}\) .
Inequality (REF ) for \(0<p\le q\le {\infty }\)  and
equality (REF ) for \(0<p\le {\infty }\)  and \(q={\infty }\) 
were proved by the author
[15]} in the following form:
\(&& \sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_q({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}\le \liminf _{n\rightarrow {\infty }}\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(T)\Vert _{L_q(Q_{\pi n}^m)}}{\Vert T\Vert _{L_p(Q_{\pi n}^m)}}, \\&&\sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_{\infty }({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}= \lim _{n\rightarrow {\infty }}\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(T)\Vert _{L_{\infty }(Q_{\pi n}^m)}}{\Vert T\Vert _{L_p(Q_{\pi n}^m)}}. \) 
The following asymptotic results
(see [15]})
are easy corollaries of (REF ) and
() (\(0<p\le q\le {\infty }\) ):
\(&& \sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_q({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}\le \liminf _{n\rightarrow {\infty }} n^{-N-m/p+m/q}\sup _{Q\in {\mathcal {T}}_{nV}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(Q)\Vert _{L_q(Q_\pi ^m)}}{\Vert Q\Vert _{L_p(Q_\pi ^m)}}, \\&&\sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_{\infty }({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}= \lim _{n\rightarrow {\infty }} n^{-N-m/p}\sup _{Q\in {\mathcal {T}}_{nV}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(Q)\Vert _{L_{\infty }(Q_\pi ^m)}}{\Vert Q\Vert _{L_p(Q_\pi ^m)}}. \) 
For \(m=1\)  and \(V=[-{\sigma },{\sigma }]\)  relations
(REF ) and () were
established by the author and Tikhonov
[68]}.
Similar relations for \(m=1,\,V=[-{\sigma },{\sigma }]\) , and
\(N=0\)  were obtained earlier by
Levin and Lubinsky [66]},
[67]}.
A special case of \(m=1,\,V=[-1,1],\,q={\infty },\,p=1\) , and
\(N=0\)  was discussed by Taikov
[63]}, [64]}
and Gorbachev [9]}.
More references can be found in
[68]}, [15]}.
Condition C3.1.1) of approximation of entire
functions of exponential type \(V\) 
by trigonometric polynomials was verified
by using univariate and multivariate Levitan's polynomials
(see [68]} for \(m=1\)  and
[15]} for \(m>1\) ).
In addition, we choose
\(B_n^\prime &=&\Bigg {\lbrace }f_n\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace :\Bigg {(}\Vert d_N(f_n)\Vert _{L_{\infty }(Q_{\pi n}^m)}=\vert d_N(f_n)(0)\vert =1\Bigg {)}\nonumber \\&\wedge &\left.\left(\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(T)\Vert _{L_{\infty }(Q_{\pi n}^m)}}{\Vert T\Vert _{L_p(Q_{\pi n}^m)}}=\Vert f_n\Vert _{L_p(Q_{\pi n}^m)}^{-1}\right)\right\rbrace ,\qquad n\in {\mathbb {N}}.\) 
Since the set \({\mathcal {T}}_{n,V}\)  consists of \(2\pi n\) -periodic
trigonometric polynomials, we have
\(\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(T)\Vert _{L_{\infty }(Q_{\pi n}^m)}}{\Vert T\Vert _{L_p(Q_{\pi n}^m)}}=\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\vert d_N(T)(0)\vert }{\Vert T\Vert _{L_p(Q_{\pi n}^m)}},\qquad n\in {\mathbb {N}}.\) 
Therefore, by (REF ) and by the standard compactness argument
(see the proof of Theorem 1.3 in [15]} for details),
we conclude that
\(B_n^\prime \ne \emptyset \)  and in addition,
\(\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(T)\Vert _{L_{\infty }(Q_{\pi n}^m)}}{\Vert T\Vert _{L_p(Q_{\pi n}^m)}}=\sup _{f_n\in B_n^\prime }\frac{\Vert d_N(f_n)\Vert _{L_{\infty }(Q_{\pi n}^m)}}{\Vert f_n\Vert _{L_p(Q_{\pi n}^m)}},\qquad n\in {\mathbb {N}}.\) 
Thus Condition C9) is satisfied for the chosen \(B_n^\prime ,\,n\in {\mathbb {N}}\) .
Next, equality (REF ) of C12.1)
holds true since for any \(f_n\in B_n^\prime \)  we have
(\(n\in {\mathbb {N}},\,s\in {\mathbb {N}}\) )
\(\left\Vert D_{n}(f_{n})\right\Vert _{F^{(1)}\left({\Omega }_{n}\right)}=\Vert d_N(f_n)\Vert _{L_{\infty }(Q_{\pi n}^m)}=\left|d_N(f_n)(0)\right|=\Vert d_N(f_n)\Vert _{L_{\infty }(Q_{\pi s}^m)}=\left\Vert D_{n}(f_{n})\right\Vert _{F^{(1)}\left({\Omega }_{s}\right)}.\) 
Further, the compactness condition of \(\lbrace B_n^\prime \rbrace _{n=1}^{\infty }\) ,
stated in (REF ) and () of C12.1)
(with a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  that is independent of \(s\) ),
is satisfied as well.
Indeed, let \(f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) . Then for \(p\in (0,{\infty })\)
\(&&\sup _{n\in {\mathbb {N}}}\Vert f_n\Vert _{L_{\infty }(Q_{\pi n}^m)}\le \sup _{n\in {\mathbb {N}}}(d_0,{\mathcal {T}}_{n,V},L_{\infty }(Q_{\pi n}^m),L_p(Q_{\pi n}^m)\Vert f_n\Vert _{L_p(Q_{\pi n}^m)}\\&&= \sup _{n\in {\mathbb {N}}}(d_0,{\mathcal {T}}_{n,V},L_{\infty }(Q_{\pi n}^m),L_p(Q_{\pi n}^m)/ (d_N,{\mathcal {T}}_{n,V},L_{\infty }(Q_{\pi n}^m),L_p(Q_{\pi n}^m)=C_8<{\infty },\) 
by the choice of \(B_n^\prime \)  and
by crude estimates of the constants in multivariate
Bernstein-Nikolskii type inequalities
(see [15]}).
Therefore,
\(\cup _{n=1}^{\infty }B_n^\prime &\subseteq &\lbrace T\in \cup _{n=1}^{\infty }{\mathcal {T}}_{n,V}:\Vert T\Vert _{L_{\infty }(Q_{\pi n}^m)}\le C_8\rbrace \nonumber \\&\subseteq &\lbrace f\in B_V:\Vert f\Vert _{L_{\infty }({\mathbb {R}}^m)}\le C_8\rbrace ,\) 
since \({\mathcal {T}}_{n,V}\subseteq B_V,\,n\in {\mathbb {N}}\) ,
and the compactness property of
the right-hand side set in (REF ) is proved
in [15]}
(a special case is discussed in
[24]}).
Note also that by [15]},
\((d_N,{\mathcal {T}}_{n,V},L_{\infty }(Q_{\pi n}^m),L_p(Q_{\pi n}^m)<{\infty },\qquad n\in {\mathbb {N}}.\) 
So (REF ) for \(0<p\le q\le {\infty }\)  and
() for \(0<p\le {\infty }\)  and \(q={\infty }\)  follow from
statements (i) and (vi) of
Corollary REF , respectively.

Example 7.3 
(Multivariate
inequalities of different metrics).
Let
\(F^{(1)}({\mathbb {R}}^m)=C_0({\mathbb {R}}^m),\,{\beta }_1=1,\,F^{(2)}({\mathbb {R}}^m)=L_p({\mathbb {R}}^m),\,{\beta }_2=\min \lbrace 1,p\rbrace ,\,B=B_V\cap L_p({\mathbb {R}}^m),\,0<p\le {\infty },\,{\Omega }_n=nV^*,\,{\Omega }_n^*=\tau nV^*,\,\tau \in (0,1),\,B_n={\mathcal {P}}_{n,m}\left|_{\Omega _n}\right.={\mathcal {P}}_{n,m},\,B_n^*={\mathcal {P}}_{n,m}\left|_{\Omega _n^*}\right.={\mathcal {P}}_{n,m}\) .
Next, let
\(d_N=\sum _{\vert {\alpha }\vert =N}b_{\alpha }D^{\alpha }\) ,
where
\(b_{\alpha }\in {\mathbb {C}}^1\)  (\({\alpha }\in {\mathbb {Z}}^m_+,\,\vert {\alpha }\vert =N\) )
and \(N\in {\mathbb {Z}}^1_+\) , are independent of
\(n\) , and let
\(D=d_N\left|_{{\mathbb {R}}^m}\right.=d_N,\,D_n=d_N\left|_{\Omega _n}\right.=d_N,\,D_n^*=d_N\left|_{\Omega _n^*}\right.=d_N,\,n\in {\mathbb {N}}\) .
Inequalities (REF ) were proved by the author
[18]} in the following form:
\(&&\limsup _{n\rightarrow {\infty }}\sup _{P\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace }\frac{\vert d_N(P)(0)\vert }{\Vert P\Vert _{L_p(nV^*)}}\le \sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\vert d_N(f)(0)\vert }{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}\nonumber \\&&= \sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_{\infty }({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}\le \liminf _{n\rightarrow {\infty }}\sup _{P\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace }\frac{\vert d_N(P)(0)\vert }{\Vert P\Vert _{L_p(\tau nV^*)}}\) 
for \(\tau \in (0,1)\) .
By letting \(\tau \rightarrow 1-\)  in (REF ),
we arrive at the following asymptotic result
for \(0<p\le {\infty }\) 
(see [18]}):
\(\sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_{\infty }({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}= \lim _{n\rightarrow {\infty }} n^{-N-m/p}\sup _{Q\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace }\frac{\vert d_N(Q)(0)\vert }{\Vert Q\Vert _{L_p(V^*)}}.\) 
For \(m=1,\,V=[-1,1]\) , \(V^*=[-1,1]\) ,
and \(d_N(f)=d^Nf/dx^N\) ,
equality (REF ) was established in
[14]}. In case when
\(V\)  and \(V^*\)  are the unit ball
\({\mathfrak {B}}_1^m\)  and \(d_{2N}=\Delta ^N\)  is the polyharmonic operator,
(REF ) was proved in [17]}
by reducing the multivariate problem to a univariate one
in weighted spaces (see Example REF ).
Condition C3.1) of approximation of entire
functions from \(B_V\cap L_p({\mathbb {R}}^m)\) 
by algebraic polynomials was verified in
[55]} and [18]}
by using approximation methods developed by Bernstein
[96]} (see also [1]}
and [98]}) and by the author
[55]}, [18]}.
In addition, we choose
\(B_n^\prime =\left\lbrace f_n\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace :\left(\vert d_N(f_n)(0)\vert =1\right)\wedge \left(\sup _{P\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace }\frac{\vert d_N(P)(0)\vert }{\Vert P\Vert _{L_p(nV^*)}}=\Vert f_n\Vert _{L_p(nV^*)}^{-1}\right)\right\rbrace ,\,\, n\in {\mathbb {N}}.\) 
By the standard compactness argument
(see the proof of Theorem 1.2 in [18]} for details),
we conclude that
\(B_n^\prime \ne \emptyset \)  and in addition,
\(\sup _{P\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace }\frac{\vert d_N(P)(0)\vert }{\Vert P\Vert _{L_p(nV^*)}}=\sup _{f_n\in B_n^\prime }\frac{\vert d_N(f_n)(0)\vert }{\Vert f_n\Vert _{L_p(nV^*)}},\qquad n\in {\mathbb {N}}.\) 
Thus Condition C9) is satisfied for the
chosen \(B_n^\prime ,\,n\in {\mathbb {N}}\) .
Next, it is easy to verify that equality (REF ) of C12.1)
holds true in this case as well
since for any \(f_n \in B_n^\prime \) 
we have
\(\left\Vert D_{n}(f_{n})\right\Vert _{F^{(1)}\left({\Omega }_{n}\right)}=\left\Vert d_N(f_n)\right\Vert _{C_0\left(nV^*\right)}=\left\Vert d_N(f_n)\right\Vert _{C_0\left(sV^*\right)}=\left\Vert D_{n}(f_{n})\right\Vert _{F^{(1)}\left({\Omega }_{s}\right)},\quad n\in {\mathbb {N}},\quad s\in {\mathbb {N}}.\) 
Further, the compactness condition of
\(\lbrace B_n^\prime \rbrace _{n=1}^{\infty }\) , stated in
(REF ) and () of C12.1)
(with a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\) 
that is independent of \(s\) ),
is satisfied. The proof of this fact given in
[18]} is based on results
of Bernstein [96]} (for \(m=1\) )
and the author [18]}
(for \(m>1\) ).
Note also that by a crude estimate of
the constant in a multivariate inequality
of different metrics
(see [18]}),
\((d_N,{\mathcal {P}}_{n,m},C_0(nV^*),L_p(nV^*)<{\infty }\)  for \(n\in {\mathbb {N}}\) .
So (REF ) for \(p\in (0,{\infty }]\)  follows from
Corollary REF  (v).

Example 7.4 
(Univariate
inequalities of different weighted metrics).
Let \(d\mu (x)=\vert x\vert ^{2\nu +1}dx,\,\nu \ge -1/2\) ,
be the univariate weighted measure on \({\mathbb {R}}^1\) . Next, let
\(F^{(1)}({\mathbb {R}}^1)=C_0({\mathbb {R}}^1),\,{\beta }_1=1,\,F^{(2)}({\mathbb {R}}^1)=L_{p,\mu }({\mathbb {R}}^1),\,{\beta }_2=\min \lbrace 1,p\rbrace \) ,
\(B=B_{1,e}\cap L_{p,\mu }({\mathbb {R}}^1):=\lbrace f\in B_{[-1,1]}:f(-x)=f(x),\,x\in {\mathbb {R}}^1\rbrace \cap L_{p,\mu }({\mathbb {R}}^1),\) 
\(0<p\le {\infty },\,{\Omega }_n=[-n,n],\,{\Omega }_n^*=[-\tau n,\tau n],\,\tau \in (0,1),\,n\in {\mathbb {N}}\) .
Next, let
\(Be({\varphi })(x):={\varphi }^{\prime \prime }(x)+((2\nu +1)/x){\varphi }^\prime (x)\)  be the Bessel operator
and let
\(D=(Be)^N\left|_{{\mathbb {R}}^m}\right.=(Be)^N,\,D_n=(Be)^N\left|_{\Omega _n}\right.=(Be)^N,\,D_n^*=(Be)^N\left|_{\Omega _n^*}\right.=(Be)^N,\,N\in {\mathbb {Z}}^1_+,\,n\in {\mathbb {N}}\) .
Further, let
\({\mathcal {P}}_{n,1,e}:=\lbrace P\in {\mathcal {P}}_{n,1}:P(-x)=P(x),\,x\in {\mathbb {R}}^1\rbrace \) 
be a subset of \({\mathcal {P}}_{n,1}\)  of all even
univariate polynomials with complex coefficients, and let
\(B_n={\mathcal {P}}_{n,1,e}\left|_{\Omega _n}\right.\)  and
\(B_n^*={\mathcal {P}}_{n,1,e}\left|_{\Omega _n^*}\right.,\,n\in {\mathbb {N}}.\)
Inequalities (REF ) were proved by the author
[17]} in the following form:
\(\lim _{n\rightarrow {\infty }}\sup _{P\in {\mathcal {P}}_{2\lfloor n/2 \rfloor ,1,e}\setminus \lbrace 0\rbrace }\frac{\vert (Be)^N(P)(0)\vert }{\Vert P\Vert _{L_{p,\mu }([-n,n])}}&\le &\sup _{f\in (B_{1,e}\cap L_{p,\mu }({\mathbb {R}}^1))\setminus \lbrace 0\rbrace }\frac{\vert (Be)^N(f)(0)\vert }{\Vert f\Vert _{L_{p,\mu }({\mathbb {R}}^1)}}\nonumber \\&\le & \lim _{n\rightarrow {\infty }}\sup _{P\in {\mathcal {P}}_{2\lfloor n/2 \rfloor ,1,e}\setminus \lbrace 0\rbrace }\frac{\vert (Be)^N(P)(0)\vert }{\Vert P\Vert _{L_{p,\mu }([-\tau n,\tau n])}}\) 
for \(\tau \in (0,1)\) .
By letting \(\tau \rightarrow 1-\)  in (REF ),
we arrive at the following asymptotic result
for \(0<p\le {\infty }\) 
(see [17]}):
\(\sup _{f\in (B_{1,e}\cap L_{p,\mu }({\mathbb {R}}^1))\setminus \lbrace 0\rbrace }\frac{\vert (Be)^N(f)(0)\vert }{\Vert f\Vert _{L_{p,\mu }({\mathbb {R}}^1)}}= \lim _{n\rightarrow {\infty }} n^{-2N-(2\nu +2)/p}\sup _{P\in {\mathcal {P}}_{2\lfloor n/2 \rfloor ,1,e}\setminus \lbrace 0\rbrace }\frac{\vert (Be)^N(P)(0)\vert }{\Vert P\Vert _{L_{p,\mu }([-1,1])}}.\) 
Verifications of Conditions C3.1)
and C12.1) in weighted spaces are provided similarly
to Example REF  (see [17]} for details).
So (REF ) for \(p\in (0,{\infty }]\)  follows from
Corollary REF  (v).

Remark 7.5 
Special cases of relations (REF ), (REF ),
and (REF )
that are discussed
in Examples
REF , REF , and REF 
and published in
[68]}, [14]}, [15]}, [17]}, [18]}
follow from statements (ii), (v), and (vi) of
Corollary REF .
In other words, the relations presented in
the aforementioned publications can be proved
by verifying the validity of the corresponding
conditions C3.1) (or C3.1.1)) and C12.1).
Despite the fact that these conditions were
not formulated explicitly in
these papers,
the proofs of
special cases of (REF ), (REF ),
and (REF )
were actually based on the verification of
these conditions.
This observation along with the general approach
to sharp constants of approximation theory
developed in [30]}, [31]} were the major
motivations for writing this paper.

Approximation of Individual Elements
(Problem B)
Here, we discuss a special case of Corollary REF  and
three examples.

General result.
We assume that all assumptions of Section REF  are held
in the following special cases.
First,
we assume that \(F^{(2)}_*=F({\mathbb {R}}^m)=F({\mathbb {R}}^m,\mu )\)  is
a \({\beta }_2\) -normed functional space
with the monotone and \({\omega }\) -continuous \({\beta }_2\) -norm
(see Definitions REF  and REF ), \({\beta }_2\in (0,1]\) , and
\(F^{(2)}_{*n}=F({\Omega }_n)\) 
or \(F^{(2)}_{*n}=F({\Omega }_n^*),\,n\in {\mathbb {N}}\) .
Next, we define the restriction operators \(L_n^{(2)}=R({\mathbb {R}}^m,{\Omega }_n^*),\,l_n^{(2)}=R({\mathbb {R}}^m,{\Omega }_n),\,n\in {\mathbb {N}}\) , and
\(l_{n,s}^{(2)}=\left\lbrace \begin{array}{ll}R({\Omega }_n,{\Omega }_s),&n\ge s,\\R({\Omega }_s,{\Omega }_n),&n< s,\end{array}\right.\quad n\in {\mathbb {N}},\quad s\in {\mathbb {N}}.\) 
We also assume that \(B,\,B_n\) , and \(B_n^*\) 
are nontrivial subspaces of \(F({\mathbb {R}}^m),\ F({\Omega }_n)\) , and \(F({\Omega }_n^*)\) ,
respectively, \(n\in {\mathbb {N}}\) .
Next, for \({\varphi }\in F({\mathbb {R}}^m)\)  we set
\({\varphi }_n:=L_n^{(2)}({\varphi })={\varphi }\left|_{\Omega _n^*}\right.\in F(\Omega _n^*)\) 
or
\({\varphi }_n:=l_n^{(2)}({\varphi })={\varphi }\left|_{\Omega _n}\right.\in F(\Omega _n)\) .
In addition, we assume that
\({\varphi }\in F({\mathbb {R}}^m)\setminus B\)  and
\({\varphi }\left|_{\Omega _n}\right.\in F(\Omega _n)\setminus B_n\)  or
\({\varphi }_n\left|_{\Omega _n^*}\right.\in F(\Omega _n^*)\setminus B_n^*\) .
Note that to the end of the example we use the
notation \({\varphi }\)  instead of
\({\varphi }\left|_{\Omega _n}\right.\)  or
\({\varphi }\left|_{\Omega _n^*}\right.\) .
The "norms" are defined by (REF ) as follows.
\(\Vert h\Vert _{F^{(2)}}:=\Vert {\varphi }-h\Vert _{F({\mathbb {R}}^m)},\quad h\in F({\mathbb {R}}^m);\qquad \Vert h_n\Vert _{F^{(2)}_n}:=\Vert {\varphi }-h_n\Vert _{F(\Omega )},\quad h_n\in F(\Omega ),\) 
where \(\Omega =\Omega _n\)  or \(\Omega =\Omega _n^*\) .
Furthermore, for \({\varphi }\in F({\mathbb {R}}^m)\)  we consider the errors of
best approximation
\(E({\varphi },B,F({\mathbb {R}}^m))\) , \(E({\varphi },B_n,F({\Omega }_n))\) , and
\(E({\varphi },B_n^*,F({\Omega }_n^*)),\,n\in {\mathbb {N}}\)  (see (REF )
or (REF )
for the definition);
by the assumptions on \({\varphi }\)  and its restriction to
\(\Omega _n\)  or \(\Omega _n^*,\,n\in {\mathbb {N}}\) ,
all these approximation errors are positive.
Finally, let
\(B_n^\prime \subseteq B_n\setminus \lbrace 0\rbrace \)  be a set
from Condition C9) such that
the following relation holds:
\(E({\varphi },B_n,F({\Omega }_n))=E({\varphi },B_n^\prime ,F({\Omega }_n)).\) 
Below we introduce some conditions on families
\(\left\lbrace B,F({\mathbb {R}}^m)\right\rbrace ,\left\lbrace B_n,F({\Omega }_n)\right\rbrace \) ,
and \(\left\lbrace B_n^*,F({\Omega }_n^*)\right\rbrace ,\linebreak n\in {\mathbb {N}}\) ,
that are needed for the validity of relations (REF )
and/or (REF ) in function spaces.

Condition C3.2*). For every \(f\in B\setminus \lbrace 0\rbrace \) 
there exists a sequence \(f_n^*\in B_n^*,\,n\in {\mathbb {N}}\) ,
such that
\(\lim _{n\rightarrow {\infty }}\left\Vert f-f_n^*\right\Vert _{F({\Omega }_n^*)}=0.\) 
Condition C10.2). (Compactness condition)
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty }\)  such that
\(f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) ,
there exist a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural
numbers and an element \(f\in B\) 
such that
\(\lim _{k\rightarrow {\infty }}\left\Vert f_{n_k}-f\right\Vert _{F\left({\Omega }_{n_k}\right)}=0.\) 
Condition C12.2). (Weak compactness condition)
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty }\) 
such that
\(f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) ,
there exists \(f\in B\)  such that
for any fixed \(s\in {\mathbb {N}}\) ,
there exists a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural numbers
(\(n_k=n_k(s),\,k\in {\mathbb {N}}\) )
such that
\(\lim _{k\rightarrow {\infty }}\left\Vert f_{n_k}-f\right\Vert _{F\left({\Omega }_{s}\right)}=0,\) 
The following corollary holds true.
Corollary 7.6 
(i) If Condition C3.2*)
is satisfied, then
\(\limsup _{n\rightarrow {\infty }} E\left({\varphi },B_n^*,F({\Omega }_n^*)\right)\le E\left({\varphi },B,F({\mathbb {R}}^m)\right).\) 
(ii) If
Condition C10.2) is satisfied, then
\(\liminf _{n\rightarrow {\infty }} E\left({\varphi },B_n,F({\Omega }_n)\right)\ge E\left({\varphi },B,F({\mathbb {R}}^m)\right).\) 
(iii) If
Condition C12.2) is satisfied,
then (REF ) holds
true.
(iv)
Let
Condition C3.2*) and either C10.2) or C12.2) be satisfied.
Then
\(\limsup _{n\rightarrow {\infty }} E\left({\varphi },B_n^*,F({\Omega }_n^*\right))\le E\left({\varphi },B,F({\mathbb {R}}^m)\right)\le \liminf _{n\rightarrow {\infty }} E\left({\varphi },B_n,F({\Omega }_n\right)).\) 

Note first that in this proof all elements in discussed
Conditions C1) through C13) are replaced without further notice
by the corresponding elements that are used in Conditions
C3.2*), C10.2), C12.2) and relations (REF ),
(REF ), (REF ). For example, in Condition
C3*) \(B_n\)  is replaced by \(B_n^*\) .
By the assumptions about the functional spaces and operators
introduced above, Conditions C4*), C6), C6b), C11), and C11a)
of Section REF  are satisfied. Indeed,
Conditions C6) and C11) are satisfied by the direct
definitions of the operators whose existence is required
in these conditions.
In addition,
Conditions C4*), C6b), and C11a) are satisfied
by Proposition REF .
For example, for every \(h\in F^{(2)}\)  we obtain by
(REF ) and by Proposition REF  (ii)
\(\liminf _{n\rightarrow {\infty }}\left\Vert l_n^{(2)}(h)\right\Vert _{ F^{(2)}_n}&=&\liminf _{n\rightarrow {\infty }}\left\Vert {\varphi }_n-l_n^{(2)}(h)\right\Vert _{ F(\Omega _n)}=\liminf _{n\rightarrow {\infty }}\left\Vert l_n^{(2)}({\varphi }-h)\right\Vert _{ F(\Omega _n)}\nonumber \\&=&\Vert {\varphi }-h\Vert _{ F({\mathbb {R}}^m)}= \Vert h\Vert _{F^{(2)}}.\) 
Therefore, Condition C6b) is satisfied. Similarly we can prove that
Conditions C4*) and C11a) are satisfied (see also Remark REF ).
Next, Conditions C3.2*), C10.2), and C12.2) of this section
are equivalent to Conditions C3*),
equation () of C10), and
equation () of C12) from Section
REF , respectively.
Therefore, statements (i), (ii), and (iii)
of Corollary REF 
follow immediately
from Corollary REF , while statement (iv) is
a direct corollary of (i), (ii), and (iii).
\(\Box \)
Remark 7.7 
Statements (i) and (ii) of Corollary REF  with
slightly different conditions were obtained in
[30]} for normed spaces and
in
[31]}
for \({\beta }_2\) -normed spaces.

Examples. There are numerous publications
[118]}, [119]}, [120]}, [96]}, [122]}, [1]}, [98]}, [125]}, [55]}, [56]}, [30]}, [31]}, [130]}, [44]} about special cases of Corollary
REF  and their modifications;
here, we call them limit theorems of approximation theory
(see also Section  B).
Examples REF , REF , and REF  below
discuss relations (REF )
proved in these papers in various situations.
Example 7.8 
(Limit theorems for multivariate periodic approximations).
Let
\(F({\mathbb {R}}^m)=L_p({\mathbb {R}}^m),\linebreak {\beta }_2=\min \lbrace 1,p\rbrace \) ,
and let \({\varphi }\in L_p({\mathbb {R}}^m)\cap M_{C,N}\) 
be a fixed function\(,\,0<p\le {\infty }\) .
Next, let
\( B=B_V\cap L_p({\mathbb {R}}^m),\,0<p\le {\infty },\,{\Omega }_n={\Omega }_n^*=Q_{{\gamma }(n)}^m,\,B_n=B_N^*={\mathcal {T}}_{n,V}\cap M_{C_9C,N+2}\left|_{Q_{{\gamma }(n)}^m}\right.={\mathcal {T}}_{n,V}\cap M_{C_9C,N+2},\,n\in {\mathbb {N}}.\) 
Here, \(C>0\)  and \(N\ge 0\)  are fixed constants;
\(C_9\)  is any constant such that \(C_9\ge C_{10}>0\) ,
and \(C_{9}\)  and \(C_{10}\)  are independent
of \(n\)  and \({\varphi }\) . In addition,
a number sequence \(\lbrace {\gamma }(n)\rbrace _{n=1}^{\infty }\)  satisfies
the conditions
\(0<{\gamma }(n)\le \pi n,\quad n\in {\mathbb {N}},\qquad \lim _{n\rightarrow {\infty }}{\gamma }(n)={\infty },\qquad \lim _{n\rightarrow {\infty }}({\gamma }(n))^{2+m/p}n^{-1/(m+2)}=0.\) 
Relations (REF ) were proved by the author [30]}, [31]}
in the following form:
\(\lim _{n\rightarrow {\infty }}E\left({\varphi },{\mathcal {T}}_{n,V}\cap M_{C_9C,N+2},L_p\left(Q_{{\gamma }(n)}^m\right)\right)=E\left({\varphi },B_V\cap L_p({\mathbb {R}}^m),L_p({\mathbb {R}}^m)\right).\) 
For \(p\in [1,{\infty }]\)  equality (REF ) was proved
in [30]} in a more general situation;
the case of \(p\in (0,1)\)  was discussed in
[31]}. For \(m=1,\,V=[-{\sigma },{\sigma }]\) , and
\(p={\infty }\) , (REF ) was established by Bernstein
[120]}.
Condition C3.2*) of approximation of entire
functions of exponential type \(V\) 
by multivariate trigonometric polynomials was verified in
[30]} (see also [15]}).
In addition, we choose
\(B_n^\prime =B_n={\mathcal {T}}_{n,V}\cap M_{C_9C,N+2}\) .
Since \({\mathcal {T}}_{n,V}\subseteq B_V\) ,
the compactness property of \(\cup _{n=1}^{\infty }B_n^\prime \) , stated in C12.2),
can be found in [139]} and
[15]}
(a special case is discussed in [24]}).
So (REF ) for \(p\in (0,{\infty }]\)  follows from Corollary
REF  (iv).

Example 7.9 
(Limit theorems for multivariate polynomial approximations).
Let
\(F({\mathbb {R}}^m)=L_p({\mathbb {R}}^m),\,{\beta }_2=\min \lbrace 1,p\rbrace ,\,B=B_V\cap L_p({\mathbb {R}}^m),\,0<p\le {\infty },\,{\Omega }_n=nV^*,\,{\Omega }_n^*=\tau nV^*,\,\tau \in (0,1),\,B_n={\mathcal {P}}_{n,m}\left|_{\Omega _n}\right.={\mathcal {P}}_{n,m},\,B_n^*={\mathcal {P}}_{n,m}\left|_{\Omega _n^*}\right.={\mathcal {P}}_{n,m},\,n\in {\mathbb {N}}.\)
Relations (REF )
were proved in
[55]}. The following result is a corollary of
(REF ):
if \({\varphi }\in L_{\infty }({\mathbb {R}}^m)\) , then
\(&&E\left({\varphi },B_{ V}\cap L_{\infty }({\mathbb {R}}^m),L_{\infty }({\mathbb {R}}^m)\right)\le \liminf _{n\rightarrow {\infty }}E\left({\varphi },{\mathcal {P}}_{n,m},L_{\infty }\left(nV^*\right)\right)\nonumber \\&&\le \limsup _{n\rightarrow {\infty }}E\left({\varphi },{\mathcal {P}}_{n,m},L_{\infty }\left(nV^*\right)\right)\le \lim _{\tau \rightarrow 1^-}E\left({\varphi },B_{\tau V}\cap L_{\infty }({\mathbb {R}}^m),L_{\infty }({\mathbb {R}}^m)\right),\) 
and if \({\varphi }\in L_p({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m),\,p\in (0,{\infty })\) , then
\(\lim _{n\rightarrow {\infty }}E\left({\varphi },{\mathcal {P}}_{n,m},L_p\left(nV^*\right)\right)=E\left({\varphi },B_{V}\cap L_p({\mathbb {R}}^m),L_p({\mathbb {R}}^m)\right).\) 
For \(p\in [1,{\infty }]\)  relations (REF ) and
(REF ) were proved
in [55]};
the case of \(p\in (0,1)\)  was discussed
in a more general and precise form in
[31]}. For \(m=1,\,V=[-{\sigma },{\sigma }]\) ,
and \(V^*=[-1/{\sigma },1/{\sigma }]\) ,
inequalities (REF ) were established by Bernstein
[96]}. Equality (REF )
for \(m=1,\,V=[-{\sigma },{\sigma }],\,V^*=[-1/{\sigma },1/{\sigma }]\) ,
and \(p\in [1,{\infty })\) 
was proved by Raitsin [125]}.
More precise relations than (REF ) were proved in
[56]}.
Condition C3.2*) of approximation of entire
functions of exponential type \(V\) 
by multivariate algebraic polynomials was verified in
[55]}
(see also [18]}).
In addition, we choose
\(B_n^\prime =\left\lbrace f_n\in {\mathcal {P}}_{n,m}:\Vert f_n\Vert _{L_p(nV^*)}\le \left(2^{{\beta }_2}+1\right)^{1/{\beta }_2}\Vert {\varphi }\Vert _{L_p({\mathbb {R}}^m)}\right\rbrace ,\qquad n\in {\mathbb {N}}.\) 
Then
\(E({\varphi },{\mathcal {P}}_{n,m},L_p(nV^*))= E({\varphi },B_n^\prime ,L_p(nV^*)),\qquad n\in {\mathbb {N}}.\) 
Indeed, given \({\varepsilon }\in \left(0,\Vert {\varphi }\Vert _{L_p({\mathbb {R}}^m)}\right)\) 
and \(n\in {\mathbb {N}}\) , let \(f_{n,{\varepsilon }}\in {\mathcal {P}}_{n,m}\)  satisfy the inequalities
\(\Vert {\varphi }\Vert _{L_p({\mathbb {R}}^m)}\ge E\left({\varphi },{\mathcal {P}}_{n,m},L_p\left(nV^*\right)\right)> \Vert {\varphi }-f_{n,{\varepsilon }}\Vert _{L_p\left(nV^*\right)}-{\varepsilon }.\) 
Then using the \({\beta }_2\) -triangle inequality,
we see that \(f_{n,{\varepsilon }}\in B_n^\prime \) ,
and, in addition,
\(E\left({\varphi },{\mathcal {P}}_{n,m},L_p\left(nV^*\right)\right)=\inf _{{\varepsilon }\in \left(0,\Vert {\varphi }\Vert _{L_p({\mathbb {R}}^m)}\right)}\Vert {\varphi }-f_{n,{\varepsilon }}\Vert _{L_p\left(nV^*\right)}\ge E\left({\varphi },B_n^\prime ,L_p\left(nV^*\right)\right).\) 
Therefore, (REF ) holds true.
The compactness property of \(\cup _{n=1}^{\infty }B_n^\prime \) , stated in C12.2),
was verified by Bernstein [96]} for \(m=1\) 
and by the author [18]} for \(m>1\) .
So (REF ) for \(p\in (0,{\infty }]\)  follows from Corollary
REF  (iv).

Example 7.10 
(Limit theorems for univariate polynomial approximations
with exponential
weight).
Let \(Q:{\mathbb {R}}^1\rightarrow [0,{\infty })\)  be an even differentiable
convex function and let \({\beta }_2\) -norms
\(\Vert \cdot \Vert _{L_{p}(W_p,{\mathbb {R}}^1)},\,p\in (0,{\infty }],\,{\beta }_2=\min \lbrace 1,p\rbrace \) ,
be defined by
(REF ) and (REF ).
In addition, let \(a_n=a_n(Q)\)  be the \(n\) th
Mhaskar-Rakhmanov-Saff number defined as the
positive root of the equation
\(n=\frac{2}{\pi }\int _0^1\frac{a_nx\,Q^\prime (a_nx)}{\sqrt{1-x^2}}dx\) 
and let
\(b_n=b_n(Q):=\frac{2}{\pi }\int _0^1\frac{Q^\prime (a_nx)\sqrt{1-x^2}}{x}dx+n/a_n.\) 
Next, let
\(F^{(2)}_*=L_p({\mathbb {R}}^1),\,B=B_{[-{\sigma },{\sigma }]}\cap L_p({\mathbb {R}}^1),\,F^{(2)}_{*n}=L_p\left(W_p(({\sigma }/b_n)\cdot ),{\mathbb {R}}^1\right),\,{\sigma }>0,\,\Omega _n=\Omega _n^*={\mathbb {R}}^1,\,B_n={\mathcal {P}}_{n,1},\,B_n^*={\mathcal {P}}_{\lfloor n/\tau \rfloor ,1},\,\tau \in (0,1),\,n\in {\mathbb {N}},\,p\in (0,{\infty }]\) .
Relations like (REF )
were proved in
[44]}. The following result is a corollary of
these relations: let \(W_p\)  satisfy some additional conditions
(see [44]});
if \({\varphi }\in L_{\infty }({\mathbb {R}}^1)\) , then
\(&&E\left({\varphi },B_{[-{\sigma },{\sigma }]}\cap L_{\infty }({\mathbb {R}}^1),L_{\infty }({\mathbb {R}}^1)\right)\nonumber \\&&\le \liminf _{n\rightarrow {\infty }}E\left({\varphi }((b_n/{\sigma })\cdot ),{\mathcal {P}}_{n,1},L_{\infty }\left(W_{\infty },{\mathbb {R}}^1\right)\right)\nonumber \\&&\le \limsup _{n\rightarrow {\infty }}E\left({\varphi }((b_n/{\sigma })\cdot ),{\mathcal {P}}_{n,1},L_{\infty }\left(W_{\infty },{\mathbb {R}}^1\right)\right)\nonumber \\&&\le \lim _{\tau \rightarrow 1^-}E\left({\varphi },B_{[-\tau {\sigma },\tau {\sigma }]}\cap L_{\infty }({\mathbb {R}}^1),L_{\infty }({\mathbb {R}}^1)\right),\) 
and if \({\varphi }\in L_p({\mathbb {R}}^1),\,p\in (0,{\infty })\) , then
\(\lim _{n\rightarrow {\infty }}(b_n/{\sigma })^{1/p}E\left({\varphi }((b_n/{\sigma })\cdot ),{\mathcal {P}}_{n,1},L_p\left(W_p,{\mathbb {R}}^1\right)\right)=E\left({\varphi },B_{[-{\sigma },{\sigma }] }\cap L_p({\mathbb {R}}^1),L_p({\mathbb {R}}^1)\right).\) 
For \(p\in (0,{\infty })\)  equality (REF ) was proved
in [44]}, while a stronger version
of relations (REF ) for \(p={\infty }\) 
was established in
[44]}. For the Hermite weight
similar results were obtained in [130]}.
Condition C3.2*) of polynomial approximation
with exponential weights of entire
functions of exponential type follows from
[44]}.
The sets \(B_n^\prime ,\,n\in {\mathbb {N}}\) , can be chosen
like in Example REF .
The compactness property of
\(\cup _{n=1}^{\infty }B_n^\prime \) , stated in C12.2),
is based on [44]}
and verified in [44]}.
So (REF ) for \(p\in (0,{\infty }]\)  follows from Corollary
REF  (iv).

Remark 7.11 
Special cases of relations (REF )
that are discussed
in Examples
REF , REF , and REF 
and published in
[96]}, [125]}, [55]}, [31]}, [44]}
follow from statement (iv) of
Corollary REF .
In other words, the relations presented in
the aforementioned publications can be proved
by verifying the validity of the corresponding
conditions C3.2*) or/and C12.2).
Despite the fact that these conditions were
not formulated explicitly in
these papers,
the proofs of
special cases of (REF )
were actually based on the verification of
these conditions.
This observation was the major
motivation for general approach developed in
[30]}, [31]}.

Approximation on Classes of Elements
(Problem C)
Here, we discuss a special case of Corollary REF  and
two examples.

General result.
We assume that all assumptions of Section REF  are held
in the following special cases.
First,
we assume that \(F^{(1)}_*=F({\mathbb {R}}^m)=F({\mathbb {R}}^m,\mu )\)  is
a \({\beta }_1\) -normed functional space
with the monotone and \({\omega }\) -continuous
\({\beta }_1\) -norm
(see Definitions REF  and REF ),
\({\beta }_1\in (0,1]\) , and
\(F^{(1)}_{*n}=F({\Omega }_n)\) 
or \(F^{(1)}_{*n}=F({\Omega }_n^*),\,n\in {\mathbb {N}}\) .
Next, we define the restriction operators \(L_n^{(1)}=R({\mathbb {R}}^m,{\Omega }_n^*),\,l_n^{(1)}=R({\mathbb {R}}^m,{\Omega }_n),\,n\in {\mathbb {N}}\) , and
\(l_{n,s}^{(1)}=\left\lbrace \begin{array}{ll}R({\Omega }_n,{\Omega }_s),&n\ge s,\\R({\Omega }_s,{\Omega }_n),&n< s,\end{array}\right.\quad n\in {\mathbb {N}},\quad s\in {\mathbb {N}}.\) 
We also assume that \(B,\,B_n\) , and \(B_n^*\) 
are nontrivial subspaces of \(F({\mathbb {R}}^m),\,F({\Omega }_n)\) , and \(F({\Omega }_n^*)\) ,
respectively, \(n\in {\mathbb {N}}\) .
In addition, let \(G\)  be a subspace of \(F({\mathbb {R}}^m),\,G_n\)  be a subspace of \(F({\Omega }_n)\) ,
and \(G_n^*\)  be a subspace of \(F({\Omega }_n^*),\,n\in {\mathbb {N}}\) .
Next, for \(f\in B,\,f_n\in B_n\) , and \(f_n^*\in B_n^*\) 
we consider the errors of
best approximation
\(E(f,G,F({\mathbb {R}}^m)),\,E(f_n,G_n,F({\Omega }_n))\) , and
\(E(f_n^*,G_n^*,F({\Omega }_n^*)),\,n\in {\mathbb {N}}\)  (see (REF )
or (REF )
for the definition).
Note that each of these
errors of
best approximation are \({\beta }_1\) -seminorms
on the corresponding spaces.
The "norms" are defined by (REF ) and ()
as follows.
\(\Vert h\Vert _{F^{(1)}}=E(h,G,F({\mathbb {R}}^m)),\quad h\in F({\mathbb {R}}^m),\quad \Vert h_n\Vert _{F^{(1)}_n}=E(h_n,\tilde{G},F(\Omega )),\quad h_n\in F(\Omega ),\) 
where \(\Omega =\Omega _n\)  or \(\Omega =\Omega _n^*\) 
and \(\tilde{G}=G_n\)  or \(\tilde{G}=G_n^*,\,n\in {\mathbb {N}}\) .
Note that \(\tilde{G}\)  in (REF ) can depend on \(h_n\) .
Finally, let
\(B_n^\prime \subseteq B_n\setminus \lbrace 0\rbrace \)  be a set
from Condition C8) such that
the following relation holds:
\(\sup _{f_n\in B_n}E(f_n,G_n,F({\Omega }_n))=\sup _{f_n\in B_n^\prime }E(f_n,G_n,F({\Omega }_n)).\) 
Below we introduce some conditions on families
\(\left\lbrace B,F({\mathbb {R}}^m)\right\rbrace ,\left\lbrace B_n,F({\Omega }_n)\right\rbrace \) ,
and \(\left\lbrace B_n^*,F({\Omega }_n^*)\right\rbrace ,\linebreak n\in {\mathbb {N}}\) ,
that are needed for the validity of relations (REF )
and/or (REF ) in function spaces.

Condition C3.3). For every \(f\in B\setminus \lbrace 0\rbrace \) 
there exists a sequence \(f_n^*\in B_n^*,\,n\in {\mathbb {N}}\) ,
such that
\(\lim _{n\rightarrow {\infty }}\left\Vert f-f_n^*\right\Vert _{F({\Omega }_n^*)}=0.\) 
Condition C4.3). For every \(h\in B\) ,
\(\liminf _{n\rightarrow {\infty }}E\left(h\left|_{\Omega _n^*}\right.,G_n^*,F({\Omega }^*_n)\right)=\liminf _{n\rightarrow {\infty }}E\left(h,G_n^*,F({\Omega }^*_n))\ge E(h,G,F({\mathbb {R}}^m)\right).\) 
Condition C6.3*a). For every \(h\in F({\mathbb {R}}^m)\) ,
\(\limsup _{n\rightarrow {\infty }}E\left(h\left|_{\Omega _n}\right.,G_n,F({\Omega }_n)\right)=\limsup _{n\rightarrow {\infty }}E\left(h,G_n,F({\Omega }_n))\le E(h,G,F({\mathbb {R}}^m)\right).\) 
Remark 7.12 
Condition C4.3) can be verified by using statements
(ii) or (iii) of Corollary REF ,
while Condition C6.3*a) can be verified by using statement
(i) of Corollary REF .

Condition C10.3*a). (Compactness condition)
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty }\)  such that
\(f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) ,
there exist a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural
numbers and an element \(f\in B\) 
such that
\(\lim _{k\rightarrow {\infty }}\left\Vert f_{n_k}-f\right\Vert _{F\left({\Omega }_{n_k}\right)}=0.\) 
The following corollary holds true.
Corollary 7.13 
(i) If Conditions C3.3) and C4.3)
are satisfied, then
\(\liminf _{n\rightarrow {\infty }} \sup _{f_n\in B_n^*}E\left(f_n,G_n^*,F({\Omega }_n^*)\right)\ge \sup _{f\in B}E\left(f,B,F({\mathbb {R}}^m)\right).\) 
(ii) Let \(\sup _{f_n\in B_n}E\left(f_n,G_n,F({\Omega }_n)\right)\)  be a
finite number for every \(n\in {\mathbb {N}}\) .
If Conditions C6.3*a) and C10.3*a) are
satisfied, then
\(\limsup _{n\rightarrow {\infty }} \sup _{f_n\in B_n}E\left(f_n,G_n,F({\Omega }_n)\right)\le \sup _{f\in B}E\left(f,B,F({\mathbb {R}}^m)\right).\) 
(iii) Let \(\sup _{f_n\in B_n}E\left(f_n,G_n,F({\Omega }_n)\right)\)  be a
finite number for every \(n\in {\mathbb {N}}\) .
If
Conditions C3.3), C4.3), C6.3*a),
and C10.3*a) are satisfied, then
\(\liminf _{n\rightarrow {\infty }} \sup _{f_n\in B_n^*}E\left(f_n,G_n^*,F({\Omega }_n^*)\right)\ge \sup _{f\in B}E\left(f,B,F({\mathbb {R}}^m)\right)\ge \limsup _{n\rightarrow {\infty }} \sup _{f_n\in B_n}E\left(f_n,G_n,F({\Omega }_n)\right).\) 

Note first that in this proof all elements in discussed
Conditions C1) through C13) and C10*a)
are replaced without further notice
by the corresponding elements that are used in Conditions
C3.3), C4.3), C6.3a), C10.3*a) and relations (REF ),
(REF ), (REF ). For example, in Condition
C3) \(B_n\)  is replaced by \(B_n^*\) .
Next, Conditions C6) and C11) of Section REF 
are satisfied by the direct
definitions of the operators whose existence is required
in these conditions.
Further, Conditions C3.3), C4.3), C6.3*a),
and C10.3*a) of this section
are equivalent to Conditions C3), C4), C6*a)
and C10*a) of Section
REF , respectively.
Therefore, statements (i) and (ii)
of Corollary REF 
follow immediately
from Corollary REF , while statement (iii) is
a direct corollary of (i) and (ii).
\(\Box \)
Remark 7.14 Statements (i) and (ii) of Corollary REF  with
slightly different conditions were obtained in
[30]} for normed spaces and
in
[31]}
for \({\beta }_1\) -normed spaces.

Examples. There is a number of publications
[169]}, [170]}, [1]}, [30]}, [31]}, [174]} about special cases of Corollary
REF  and their modifications;
here, we call them
relations between constants of approximation theory
(see also Section  C).
Examples REF  and REF  below
discuss relations like (REF )
proved in these papers in various situations.
Example 7.15 
(Periodic approximation on \(H_{\omega }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },n,m})\) ).
Let
\(F({\mathbb {R}}^m)=L_{\infty }({\mathbb {R}}^m),\,{\beta }_1=1,\,B=H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }(R^m),\,B_n^*=H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m},\,G=B_V\cap L_{\infty }({\mathbb {R}}^m),\,G_n^*={\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},\,{\Omega }_n^*=Q_{\pi n/2}^m,\,n\in {\mathbb {N}}.\)
Let us choose \(M=M(n)=\pi n/2,\,n\in {\mathbb {N}}\) ,
and let \(f\in H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }(R^m)\) .
Then by Lemma REF , there exists a function
\(f_n^*=\tilde{f}:{\mathbb {R}}^m\rightarrow {\mathbb {R}}^m\) 
such that \(f_n^*=f\)  on \(Q_{\pi n/2}^m\)  and
\(f_n^*\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}\) .
Since \(f-f_n^*=0\)  on \(Q_{\pi n/2}^m\) ,
Condition C3.3) is satisfied.
Next, Condition C4.3) is satisfied as well. Indeed,
let a number sequence \(\lbrace {\gamma }(n)\rbrace _{n=1}^{\infty }\) 
satisfy (REF ).
Then
for all large enough \(n,\,{\gamma }(n)\le \pi n/2\) ,
since by (REF ), \({\gamma }(n)=o(n)\)  as \(n\rightarrow {\infty }\) .
In addition, for any
\(h\in H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)\) ,
we have \(h\in M_{{\omega }(2),1} \cap L_{\infty }({\mathbb {R}}^m)\) ,
by (REF ).
Then using relation (REF ) for \(p={\infty },\,C={\omega }(2)\) ,
and \(N=1\) ,
we obtain
\(&& \liminf _{n\rightarrow {\infty }}E\left(h,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{\pi n/2)}^m\right)\right)\\&&\ge \lim _{n\rightarrow {\infty }}E\left(h,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{{\gamma }(n))}^m\right)\right)\\&& = E\left(h,B_V\cap L_{\infty }({\mathbb {R}}^m),L_{\infty }({\mathbb {R}}^m)\right),\qquad h\in H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m).\) 
Thus Condition C4.3) is satisfied.
We recall that \(C_9\)  is the
same constant as in Example
REF , i.e., \(C_9\) 
is any constant such that \(C_9\ge C_{10}>0\) ,
and \(C_9\)  and \(C_{10}\)  are independent of \(n\)  and \(h\) .
Therefore using Corollary REF  (i), we obtain
\(&&\liminf _{n\rightarrow {\infty }}\sup _{f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&&\ge \sup _{f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{\pi n/2}^m\right)\right)\nonumber \\&& \ge \sup _{f\in H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m) ,L_{\infty }\left({\mathbb {R}}^m\right)\right).\) 
Furthermore, we prove that for any
\(f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m},\,n\in {\mathbb {N}}\) , and for large enough \(C_9\ge C_{10}\) ,
the following equality holds true:
\(E\left(f_n,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{\pi n}^m\right)\right)= E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right).\) 
Indeed, let us set
\({\mathcal {T}}_{n,V,1}:=\left\lbrace g_n\in {\mathcal {T}}_{n,V}:E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\ge \Vert f_n-g_n\Vert _{L_{\infty }\left(Q_{\pi n}^m\right)}-1\right\rbrace .\) 
If \(g_n\in {\mathcal {T}}_{n,V,1}\) , then by (REF ),
\(\vert g_n(x)\vert &\le & E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)+1+\vert f_n(x)\vert \nonumber \\&\le & E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)+1+{\omega }(2)(1+\vert x\vert ),\qquad x\in {\mathbb {R}}^m.\) 
Since by Jackson's theorem
(see [24]}),
\(&& \sup _{n\in {\mathbb {N}}}\sup _{f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&&=\sup _{n\in {\mathbb {N}}}\sup _{f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n(n\cdot ),{\mathcal {T}}_{nV},L_{\infty }\left(Q_{\pi }^m\right)\right)<{\infty },\) 
we see from (REF ) and (REF ) that
\({\mathcal {T}}_{n,V,1}\subseteq M_{C_{11},1}\) , where
\(C_{11}\)  is independent of \(n\) .
Then setting
\(C_9=\max \lbrace C_{10},C_{11}/{\omega }(2)\rbrace \) ,
we obtain
\(&& E\left(f_n,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{\pi n}^m\right)\right)\le E\left(f_n,{\mathcal {T}}_{n,V}\cap M_{C_{11},1},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&& \le E\left(f_n,{\mathcal {T}}_{n,V,1},L_{\infty }\left(Q_{\pi n}^m\right)\right)= E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right).\) 
Thus (REF ) follows from (REF ).
Finally combining (REF ) and (REF ),
we obtain
\(&&\liminf _{n\rightarrow {\infty }}\sup _{f_n\in H_{{\omega }}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&&=\liminf _{n\rightarrow {\infty }}\sup _{f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&& \ge \sup _{f\in H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m) ,L_{\infty }\left({\mathbb {R}}^m\right)\right)\nonumber \\&& =\sup _{f\in H_{{\omega }}({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m) ,L_{\infty }\left({\mathbb {R}}^m\right)\right).\) 
Next, by Lemma REF , for any
\(f_n\in H_{\omega }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}\) ,
\(E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)=E\left(f_n,B_V\cap L_{\infty }({\mathbb {R}}^m) ,L_{\infty }\left({\mathbb {R}}^m\right)\right),\qquad n\in {\mathbb {N}}.\) 
Taking into account a trivial inclusion
\(H_{\omega }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}\subseteq H_{\omega }({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)\) ,
we obtain from (REF )
\(&&\limsup _{n\rightarrow {\infty }}\sup _{f_n\in H_{\omega }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&& \le \sup _{f\in H_{\omega }({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m) ,L_{\infty }\left({\mathbb {R}}^m\right)\right).\) 
Combining (REF ) and (REF ), we arrive at
\(&&\lim _{n\rightarrow {\infty }}\sup _{f_n\in H_{\omega }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&& = \sup _{f\in H_{\omega }({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m),L_{\infty }\left({\mathbb {R}}^m\right)\right).\) 
The corresponding values of the sharp constants
in (REF ) were found only for \(m=1\) 
and for a concave modulus of continuity \({\omega }\) 
(see [176]}, [177]}).
For the Hölder class \(H^{\lambda }({\mathbb {R}}^m)=H_{\omega }({\mathbb {R}}^m)\)  with
\({\omega }(\tau )=\tau ^{\lambda },\,\tau \in (0,{\infty }),\,0<{\lambda }\le 1\) ,
(REF ) immediately implies the relation
\(&&\lim _{n\rightarrow {\infty }} n^{\lambda }\sup _{f\in H^{\lambda }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi ,m}}E\left(f,{\mathcal {T}}_{nV},L_{\infty }\left(Q_{\pi }^m\right)\right)\nonumber \\&& = \sup _{f\in H^{\lambda }({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m),L_{\infty }\left({\mathbb {R}}^m\right)\right).\) 
A more precise result with
\(H^{\lambda }({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)\)  replaced by \(H^{\lambda }({\mathbb {R}}^m)\) 
was obtained in [30]}.
The corresponding univariate result was proved by Bernstein
[169]}. Bernstein [170]} attempted
to extend this result to the class \(W^sH^{\lambda }({\mathbb {R}}^1)\) 
but his proof was incomplete (see also Example REF ).

Example 7.16 
(Periodic approximation on \(W^sH_{\omega }({\mathbb {R}}^1)\) ).

The author [174]} solved
a relatively long-standing problem of finding
the exact value of
\(\sup _{f\in W^sH_{\omega }({\mathbb {R}}^1)}E(f,B_{[-{\sigma },{\sigma }]},L_{\infty }({\mathbb {R}}^1)),\,s\in Z_+^1\) ,
where \({\omega }\)  is a concave modulus of continuity.
The proof is based on a solution of the corresponding
periodic problem (see Korneichuk [182]},
[176]}) and on a limit relation
between periodic and nonperiodic approximations
like (REF ) (see [174]}).
Note that for \(s=0\)  the problem was solved by
Dzyadyk [177]}. More references can be found
in [174]}.
